The radical sign \sqrt{\phantom{x}} means the principal (non-negative) root, so the answer is 9, not -9.
Answer: 9
Check:9^2 = 81 ✓.
Tip: Memorize the perfect squares from 1^2 = 1 to 15^2 = 225. Then every square root of a perfect square is instant — and you will not confuse "square root" with "divide by 2".
Question 2 of 36
No calculatorFill in
Evaluate the expression.
(-7)^2 - 3^2
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The base of (-7)^2 is -7, including the negative sign. What is a negative times a negative?
Step-by-step solution
Square the first term. The parentheses mean the base is -7: (-7)^2 = (-7) \times (-7) = 49 (negative × negative = positive).
Square the second term: 3^2 = 3 \times 3 = 9.
Subtract: 49 - 9 = 40.
Answer: 40
Check: On the TI-30XS: ((−)7)x²−3x²enter shows 40 ✓.
Tip: Squaring is not doubling. (-7)^2 is not -14 and 3^2 is not 6; if you got -14 - 6 = -20, you doubled. And if you got -49 - 9 = -58, you forgot that the parentheses put the negative sign inside the square.
Question 3 of 36
No calculator
A square garden has an area of 144 square feet. What is the length of one side of the garden?
Hint
The area of a square is side × side. Which number times itself is 144?
Step-by-step solution
The area of a square is A = s^2 (side times side).
Here s^2 = 144, so s = \sqrt{144}.
12 \times 12 = 144, so s = 12. A length cannot be negative, so only +12 makes sense.
Tip: Going from area back to side always means taking a square root. Dividing by 4 finds a side from the perimeter, not the area.
Question 4 of 36
No calculator
Find \sqrt{121} - \sqrt{25}.
Hint
Find each square root on its own first. Which number times itself gives 121? Which gives 25?
Step-by-step solution
\sqrt{121} = 11 because 11 \times 11 = 121.
\sqrt{25} = 5 because 5 \times 5 = 25.
Subtract: 11 - 5 = 6.
Answer: 6
Check: Square each root back: 11^2 = 121 ✓ and 5^2 = 25 ✓. Notice that \sqrt{121 - 25} = \sqrt{96} \approx 9.8, which is not 6 — you cannot subtract under the root first.
Tip: Memorize the perfect squares up to 15^2 = 225: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. They come up all the time on the no-calculator part of the GED.
Question 5 of 36
No calculatorFill in
Compute the value of 15^2 - \sqrt{169}.
Type your answer in the box. Fractions like 3/4 are fine.
Hint
15^2 means 15 \times 15, not 15 \times 2. For \sqrt{169}, think of a number that times itself gives 169.
Square root:\sqrt{169} = 13 because 13 \times 13 = 169.
Subtract:225 - 13 = 212.
Answer: 212
Check:13^2 = 169 ✓, and 212 + 13 = 225 = 15^2 ✓.
Tip: The most common slip is reading 15^2 as 15 \times 2 = 30. The small 2 means "multiply the number by itself". Squaring and square-rooting undo each other: \sqrt{15^2} = 15.
Question 6 of 36
No calculator
A square city park covers an area of 2,500 square yards. How long is one side of the park?
Hint
For a square, area = side × side. Which number times itself gives 2,500?
Step-by-step solution
A square's area is A = s^2, so the side is s = \sqrt{A}.
s = \sqrt{2{,}500}.
Split off the zeros in pairs: 2{,}500 = 25 \times 100, and \sqrt{25} = 5, \sqrt{100} = 10. So \sqrt{2{,}500} = 5 \times 10 = 50.
Answer: 50 yards
Check:50 \times 50 = 2{,}500 square yards ✓.
Tip: For perfect squares that end in zeros, each pair of zeros under the root becomes one zero in the answer: \sqrt{2{,}500} = 50, \sqrt{40{,}000} = 200.
Question 7 of 36
No calculator
Which whole number is closest to \sqrt{90}?
Hint
Find the two perfect squares on either side of 90. Which one is 90 closer to?
Step-by-step solution
Perfect squares near 90: 9^2 = 81 and 10^2 = 100.
Since 81 < 90 < 100, we know 9 < \sqrt{90} < 10.
Distance from 90 to 81 is 9; distance from 90 to 100 is 10. So 90 is slightly closer to 81, and \sqrt{90} is closer to 9.
More precisely, \sqrt{90} \approx 9.49, which rounds to 9.
Answer: 9
Check:9.5^2 = 90.25. Since 90 < 90.25, \sqrt{90} is just below 9.5, so it rounds to 9 ✓.
Tip: To decide which integer is closest, square the halfway number (like 9.5^2 = 90.25). If your number is below that, round down; above it, round up. Comparing distances (9 vs 10) is a good first guess, but the halfway test is exact.
Question 8 of 36
No calculator
Simplify \sqrt{\dfrac{16}{81}}.
Hint
Take the square root of the numerator and the square root of the denominator separately.
Step-by-step solution
The square root of a fraction is the root of the top over the root of the bottom: \sqrt{\dfrac{16}{81}} = \dfrac{\sqrt{16}}{\sqrt{81}}.
Tip: The square root of a proper fraction is bigger than the fraction itself (\tfrac{4}{9} \approx 0.44 > \tfrac{16}{81} \approx 0.20). If your answer got smaller, you probably squared instead of rooting.
Question 9 of 36
No calculator
What is the value of -5^2 + (-5)^2?
Hint
In -5^2 the exponent applies only to the 5. In (-5)^2 it applies to -5.
Step-by-step solution
-5^2 means "the opposite of 5^2". The exponent touches only the 5: -5^2 = -(5 \times 5) = -25.
(-5)^2 means (-5) \times (-5) = 25. The parentheses make -5 the base.
Add: -25 + 25 = 0.
Answer: 0
Check: On the TI-30XS, (−)5x² gives -25 and ((−)5)x² gives 25; -25 + 25 = 0 ✓.
Tip: Read exponents carefully: the exponent applies only to the symbol directly in front of it. If the negative sign should be squared too, there must be parentheses.
Question 10 of 36
No calculator
Between which two consecutive whole numbers is \sqrt{58}?
Hint
Find the perfect squares just below and just above 58.
Step-by-step solution
List perfect squares near 58: 7^2 = 49 and 8^2 = 64.
Since 49 < 58 < 64, taking square roots keeps the order: \sqrt{49} < \sqrt{58} < \sqrt{64}.
So 7 < \sqrt{58} < 8.
Answer: between 7 and 8
Check: The calculator gives \sqrt{58} \approx 7.62, which is between 7 and 8 ✓.
Tip: Build a "perfect-square fence" around the number: the closest perfect square below and the closest above. Their roots are the two whole numbers you need — no calculator required.
Question 11 of 36
No calculatorFill in
What is the value of \sqrt{\dfrac{49}{64}}?
Enter your answer as a fraction or a decimal.
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The square root of a fraction is the square root of the numerator over the square root of the denominator.
Step-by-step solution
Split the root over the fraction: \sqrt{\dfrac{49}{64}} = \dfrac{\sqrt{49}}{\sqrt{64}}.
\sqrt{49} = 7 because 7 \times 7 = 49.
\sqrt{64} = 8 because 8 \times 8 = 64.
So the value is \dfrac{7}{8}, which is 0.875 as a decimal.
Tip: Roots (and squares) work on the top and bottom of a fraction separately. Note that \tfrac{7}{8} is bigger than \tfrac{49}{64} \approx 0.77: the square root of a number between 0 and 1 is larger than the number itself.
Question 12 of 36
No calculatorHot spot
Click the point on the number line that best shows the location of \sqrt{40}.
Click on the figure to mark your answer (or focus it and use the arrow keys).
Hint
Which two perfect squares is 40 between? Is 40 closer to the smaller one or the larger one?
Step-by-step solution
Find the perfect squares around 40: 6^2 = 36 and 7^2 = 49.
Since 36 < 40 < 49, we know 6 < \sqrt{40} < 7.
Decide where between 6 and 7: 40 is 4 above 36 but 9 below 49, so \sqrt{40} is closer to 6 than to 7 — roughly a third of the way from 6 to 7.
Calculator check: \sqrt{40} \approx 6.32.
Answer: the point a little to the right of 6, at about 6.3.
Check:6.3^2 = 39.69 and 6.4^2 = 40.96, so \sqrt{40} is between 6.3 and 6.4 ✓.
Tip: Do not click 4 (40 \div 10) or 7 (rounding up). First build the perfect-square fence, then decide which side of the fence the number leans toward.
Question 13 of 36
A square rug covers an area of 50 square feet. To the nearest tenth of a foot, how long is each side of the rug?
Hint
Side of a square = square root of its area. Round only at the end.
Step-by-step solution
Area of a square: A = s^2, so s = \sqrt{A} = \sqrt{50}.
Estimate first: 7^2 = 49 and 8^2 = 64, so \sqrt{50} is just a little more than 7.
Round to the nearest tenth: the hundredths digit is 7, so round the tenths digit up: 7.1.
Answer: about 7.1 feet
Check:7.1^2 = 50.41, very close to 50 ✓ (and 7.0^2 = 49 is farther away).
Tip: Keep the exact value until the last step. Rounding to the "nearest perfect square" (49 → 7) gives 7.0, which is not accurate enough when the question asks for tenths.
Question 14 of 36
No calculatorDrop-down
Select the expression that makes the statement true.
In simplest form, \sqrt{72} is equal to .
Choose an answer from each menu.
Hint
Find the largest perfect square that divides 72 evenly.
Step-by-step solution
List perfect squares that divide 72: 4 (72 = 4 \times 18), 9 (72 = 9 \times 8), and 36 (72 = 36 \times 2). The largest is 36.
Tip: Only the perfect-square factor comes out, and it comes out as its root (36 comes out as 6). Products can be split under a root; sums cannot.
Question 15 of 36
No calculatorHot spot
On the number line below, select the point that marks \sqrt{20} most accurately.
Click on the figure to mark your answer (or focus it and use the arrow keys).
Hint
Which two perfect squares is 20 between? Is 20 nearer the smaller one or the larger one?
Step-by-step solution
Perfect squares around 20: 4^2 = 16 and 5^2 = 25.
Since 16 < 20 < 25, we know 4 < \sqrt{20} < 5.
Where between 4 and 5? 20 is 4 above 16 and 5 below 25, so it sits a bit less than halfway. Check the midpoint: 4.5^2 = 20.25, which is just above 20. So \sqrt{20} is just below 4.5.
More precisely, \sqrt{20} \approx 4.47. Click just to the left of the 4.5 tick mark.
Answer: the point at about 4.47 (just left of 4.5)
Check:4.47^2 = 19.98, very close to 20 ✓.
Tip: On number-line questions, first bracket the root between two whole numbers, then use the halfway square (like 4.5^2 = 20.25) to decide which half it falls in.
Question 16 of 36
A solar panel is a perfect square and covers 18 square feet of roof. Rounded to the nearest tenth of a foot, what is the length of one edge?
Hint
Side = square root of the area. Use the calculator, then round to one decimal place.
Step-by-step solution
For a square, s = \sqrt{A} = \sqrt{18}.
18 is not a perfect square. It is between 16 and 25, so the side is between 4 and 5 feet — and close to 4.
Calculator: \sqrt{18} \approx 4.2426.
Round to the nearest tenth: the hundredths digit is 4 (less than 5), so round down to 4.2.
Answer: about 4.2 feet
Check:4.2^2 = 17.64 and 4.3^2 = 18.49. 18 is closer to 17.64 (0.36 away) than to 18.49 (0.49 away), so 4.2 is the better tenth ✓.
Tip: On the TI-30XS, press 2nd then x² to get the \sqrt{\ } symbol, type 18, press the right arrow to leave the root, and press enter. The screen shows 3\sqrt{2}; press toggle to see 4.2426\ldots Always do a quick bracket estimate (here 4 to 5) so you notice a typing slip.
Question 17 of 36
No calculatorFill in
If x^2 + 19 = 100 and x is greater than 0, what is x?
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Get x^2 by itself first. Then undo the square with a square root.
Step-by-step solution
Undo "+ 19" by subtracting 19 from both sides: x^2 = 100 - 19 = 81.
Undo the square by taking the square root: x = \sqrt{81} or x = -\sqrt{81}.
\sqrt{81} = 9, so x = 9 or x = -9.
The question asks for the positive value: x = 9.
Answer: 9
Check:9^2 + 19 = 81 + 19 = 100 ✓. (Also (-9)^2 + 19 = 100, which is why the question says "positive".)
Tip: Isolate the squared term before taking the root. A common mistake is \sqrt{100} - 19 = -9 — that undoes the steps in the wrong order.
Question 18 of 36
No calculator
Which expression shows \sqrt{48} in simplest form?
Hint
Look for the largest perfect square that divides 48: 4? 9? 16? 25?
Step-by-step solution
List perfect squares that divide 48: 4 (48 = 4 \times 12) and 16 (48 = 16 \times 3). The largest is 16.
Tip: If you use a smaller square first (like 4), you get 2\sqrt{12} — not finished, because 12 still contains the square 4. Using the largest square factor gets you there in one step.
Question 19 of 36
A square bedroom has a floor area of 169 square feet. Baseboard trim will go around the whole room except a 3-foot doorway. Trim costs $1.80 per foot. How much will the trim cost?
Hint
Find the side length from the area. Then find the distance around the room, remove the doorway, and multiply by the price.
Tip: Read every condition in a multi-step problem. Words like "except", "only", or "both" change one step. Writing the unit next to every number (sq ft vs ft vs $) keeps you from using the area where a length belongs.
Question 20 of 36
No calculatorSelect all
Which of these statements about square roots are true? Select all that apply.
Select all that apply.
Hint
Compute both sides of each statement. Remember: a root can be split over multiplication, but not over addition or subtraction.
Check: For statement 6, use decimals: 1.414 + 1.414 = 2.828 \ne 2 ✓.
Tip: Product rule: \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} — always works (for non-negative numbers). Sum rule: \sqrt{a + b} \ne \sqrt{a} + \sqrt{b} — never assume it.
Question 21 of 36
No calculatorDrop-down
Select the correct answer from each drop-down menu.
(-3)^2 \cdot (-2) is equal to , and -(3 \cdot 2)^2 is equal to .
Choose an answer from each menu.
Hint
Do what is inside parentheses first, then the exponent, then multiply. A minus sign outside the parentheses is applied last.
Step-by-step solution
First expression: (-3)^2 \cdot (-2)
Exponent first: (-3)^2 = (-3)(-3) = 9.
Multiply: 9 \cdot (-2) = -18.
Second expression: -(3 \cdot 2)^2
3. Parentheses first: 3 \cdot 2 = 6.
4. Exponent: 6^2 = 36.
5. The minus sign in front means "the opposite of": -36.
Answer:-18 and -36
Check: In the first expression the square only applies to -3 (inside its parentheses), so the -2 stays negative ✓. In the second the negative sign is not inside the parentheses, so it is not squared ✓.
Tip: Ask "what exactly is the exponent touching?" If the negative sign is inside the parentheses with the base, it gets squared. If it is outside, it stays and is applied at the end.
Question 22 of 36
No calculator
On a number line, -\sqrt{150} lies between two consecutive integers. Which two?
Hint
First place \sqrt{150} between two whole numbers using perfect squares. Then think about what the negative sign does.
Step-by-step solution
Perfect squares around 150: 12^2 = 144 and 13^2 = 169.
Since 144 < 150 < 169, we have 12 < \sqrt{150} < 13 (about 12.25).
Taking the opposite flips the order: -13 < -\sqrt{150} < -12.
So -\sqrt{150} \approx -12.25, which lies between -13 and -12.
Answer: between -13 and -12
Check:(-12.25) is to the left of -12 and to the right of -13 on a number line ✓. And 12.25^2 = 150.06 \approx 150 ✓.
Tip: The negative sign is outside the root, so -\sqrt{150} is a real number (just negative). Find the root of the positive number first, then attach the sign.
Question 23 of 36
No calculatorFill in
A square patio is covered with 324 square stone tiles. Each tile is 1 foot by 1 foot, and the tiles form a perfect square with no gaps. How many tiles are along one edge of the patio?
tiles
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The tiles form a square: (tiles per edge) × (tiles per edge) = 324.
Step-by-step solution
The patio is an n \times n square of tiles, so n^2 = 324 and n = \sqrt{324}.
Bracket: 10^2 = 100 and 20^2 = 400, so n is between 10 and 20 — closer to 20.
The last digit of 324 is 4, so n ends in 2 or 8 (since 2^2 = 4 and 8^2 = 64). Try 18: 18^2 = 324 ✓.
Tip: Without a calculator, use the last digit to narrow down a perfect square root: squares ending in 4 come from numbers ending in 2 or 8; ending in 9 from 3 or 7; ending in 6 from 4 or 6; ending in 1 from 1 or 9.
Question 24 of 36
A square patio has an area of 225 square feet. A homeowner wants to put a decorative border along all four edges. The border costs $12.50 per foot. How much will the border cost?
Hint
First find the side from the area, then the distance around the patio, then the cost.
Step-by-step solution
Side:A = s^2 = 225, so s = \sqrt{225} = 15 feet.
Perimeter (all four edges): 4 \times 15 = 60 feet.
Cost:60 \times \$12.50 = \$750.00.
Answer: $750.00
Check: Work backwards: \$750 \div \$12.50 = 60 ft of border; 60 \div 4 = 15 ft per side; 15^2 = 225 sq ft ✓.
Tip: Multi-step area problems usually need three moves: area → side (square root), side → perimeter (×4), perimeter → cost (× price). Write each result down with its unit so you know which number to use next.
Question 25 of 36
No calculatorFill in
Evaluate the expression.
\sqrt{0.36} + \sqrt{1.44}
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Write each decimal as a fraction over 100. Then take the square root of the top and bottom.
Tip: A square root has half as many decimal places as the number under the root (0.36 has two places, so its root has one: 0.6). The tempting answer 0.06 + 0.12 = 0.18 fails the check, since 0.06^2 = 0.0036.
Question 26 of 36
No calculatorHot spot
The number line runs from 0 to 12. Click where the value of the sum \sqrt{50} + 3 falls.
Click on the figure to mark your answer (or focus it and use the arrow keys).
Hint
Estimate \sqrt{50} first (which two perfect squares is 50 between?). Then add 3 to your estimate.
Step-by-step solution
Estimate \sqrt{50}: 7^2 = 49 and 8^2 = 64, so 7 < \sqrt{50} < 8.
50 is only 1 more than 49, so \sqrt{50} is just a little above 7 (about 7.07).
Tip: The 3 is not under the root. \sqrt{50} + 3 is very different from \sqrt{53} \approx 7.28. Estimate the root first, then do the arithmetic outside it.
Question 27 of 36
No calculatorFill in
Evaluate the expression.
\sqrt{0.81} \times \sqrt{\frac{25}{4}}
Enter your answer as a decimal or a fraction.
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Find each root separately. For the decimal, ask which decimal times itself gives 0.81. For the fraction, root the top and the bottom.
Step-by-step solution
\sqrt{0.81} = 0.9 because 0.9 \times 0.9 = 0.81. (Think: 9 \times 9 = 81, and one decimal place times one decimal place gives two.)
Check: Combine first: 0.81 \times \tfrac{25}{4} = \tfrac{20.25}{4} = 5.0625, and \sqrt{5.0625} = 2.25 because 2.25^2 = 5.0625 ✓.
Tip: The square root of a decimal has half as many decimal places: \sqrt{0.81} = 0.9, \sqrt{0.0081} = 0.09. A common error is \sqrt{0.81} = 0.09.
Question 28 of 36
Kim is building a square vegetable bed with an area of 60 square feet. She will put fencing around all four sides. Fencing is sold only in whole feet. What is the least number of feet of fencing she must buy?
Hint
Find the side with a square root (keep the decimals), find the perimeter, and only then round. Should you round up or down when buying material?
Step-by-step solution
Side:s = \sqrt{60} \approx 7.746 feet. Keep all the decimals for now.
Perimeter:4 \times 7.746 \approx 30.98 feet.
She needs at least 30.98 feet. Fencing is sold in whole feet, so 30 feet would be too short. Round up: 31 feet.
Answer: 31 feet
Check: 31 feet covers 31 \div 4 = 7.75 feet per side, and 7.75^2 \approx 60.06 \ge 60 ✓. With 30 feet, each side would be 7.5 feet and the area only 56.25 square feet ✗.
Tip: Round only at the end. Rounding the side to 7 feet first would give 28 feet — almost 3 feet short. When buying enough material, always round up.
Question 29 of 36
No calculator
Which value or values of x make the equation x^2 = 0.64 true?
Hint
Which decimal times itself gives 0.64? Could a negative number squared also give 0.64?
Step-by-step solution
Look for a number that times itself gives 0.64. Since 8 \times 8 = 64, try 0.8: 0.8 \times 0.8 = 0.64 ✓.
A negative number squared is positive too: (-0.8) \times (-0.8) = 0.64 ✓.
So the equation x^2 = 0.64 has two solutions: x = 0.8 and x = -0.8.
Answer: 0.8 and -0.8
Check:0.8^2 = 0.64 ✓ and (-0.8)^2 = 0.64 ✓. Also 0.08^2 = 0.0064, which is too small ✓.
Tip: The symbol \sqrt{0.64} means only the positive root, 0.8. But an equationx^2 = k (with k > 0) has two answers, \pm\sqrt{k}. Read whether the question gives you a root symbol or an equation.
Question 30 of 36
No calculatorDrop-down
Select the correct answer from each drop-down menu.
In simplest form, \sqrt{200} is equal to , and its value is between .
Choose an answer from each menu.
Hint
Find the largest perfect square that divides 200. For the estimate, which perfect squares is 200 between?
Step-by-step solution
Simplest form
Perfect squares that divide 200: 4, 25 and 100. The largest is 100.
Estimate
3. 14^2 = 196 and 15^2 = 225. Since 196 < 200 < 225, \sqrt{200} is between 14 and 15.
4. Check with the radical form: 10\sqrt{2} \approx 10 \times 1.414 = 14.14 ✓.
Answer:10\sqrt{2}; between 14 and 15
Check:(10\sqrt{2})^2 = 100 \times 2 = 200 ✓.
Tip: Simplest radical form and a decimal estimate are two views of the same number. Use one to check the other: 10\sqrt{2} must be about 14.1, and 14.1^2 \approx 200.
Question 31 of 36
No calculator
A square sign has an area of 20.25 square feet. A designer makes a new square sign whose sides are each 1.5 feet longer. What is the area of the new sign?
Hint
Find the side of the original sign first. Add 1.5 feet to the side — not to the area — then square.
Check: Draw it: the new square is the old 4.5 \times 4.5 square plus two 4.5 \times 1.5 strips plus a 1.5 \times 1.5 corner. 20.25 + 6.75 + 6.75 + 2.25 = 36 ✓.
Tip: Changes to a square's side must go through the side. You cannot add the change to the area, and (a + b)^2 is nota^2 + b^2 — it is missing the two middle strips.
Question 32 of 36
Fill in
A community center wants a square banner with an area of 30 square feet. How long should each side be, rounded to the nearest hundredth of a foot?
feet
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Side = square root of the area. Use the calculator and round to two decimal places.
Step-by-step solution
For a square, s = \sqrt{A} = \sqrt{30}.
Bracket: 5^2 = 25 and 6^2 = 36, so the side is between 5 and 6 feet — a bit below the middle.
Calculator: \sqrt{30} \approx 5.47722\ldots
Round to the nearest hundredth: look at the thousandths digit, 7. Since 7 \ge 5, round the hundredths up: 5.48.
Answer: about 5.48 feet
Check:5.48^2 = 30.0304 and 5.47^2 = 29.9209. 30 is closer to 30.0304 (0.03 away) than to 29.9209 (0.08 away), so 5.48 is right ✓.
Tip: On the TI-30XS: 2ndx² (for \sqrt{\ }), type 30, press enter. If the screen shows \sqrt{30} instead of a decimal, press the toggle key to switch to the decimal form.
Question 33 of 36
No calculatorSelect all
Select all of the statements that are true.
Select all that apply.
Hint
Test each statement by calculating both sides. Remember what the \sqrt{\phantom{x}} sign means and what happens to negative numbers.
\sqrt{0.25} = 0.5 because 0.5 \times 0.5 = 0.25. True.
x^2 = 49: both 7^2 = 49 and (-7)^2 = 49, so the solutions are 7 and -7. True.
\sqrt{-49}: no real number times itself is negative, so it is not a real number; it is not -7 (since (-7)^2 = 49, not -49). False.
Answer: the true statements are \sqrt{4 \cdot 25} = \sqrt{4} \cdot \sqrt{25}; \sqrt{0.25} = 0.5; and x^2 = 49 has solutions 7 and -7.
Check: Every true statement survives a direct numeric test; every false one fails it ✓.
Tip: Square roots split over multiplication and division, never over addition or subtraction. And keep two ideas apart: the equation x^2 = 49 has two answers, but the symbol \sqrt{49} means only +7.
Question 34 of 36
No calculator
Which expression is not equal to 12?
Hint
Evaluate each one. Which rule lets you split a root, and which does not?
\sqrt{64} + \sqrt{80} = 8 + \sqrt{80}. Since 8^2 = 64 < 80 < 81 = 9^2, \sqrt{80} \approx 8.94, so the sum is about 16.94. Not equal to 12.
The trap: 64 + 80 = 144 and \sqrt{144} = 12, so it is tempting to think \sqrt{64} + \sqrt{80} = \sqrt{144}. But a root of a sum is not the sum of the roots.
Answer: \sqrt{64} + \sqrt{80}
Check:\sqrt{64} = 8 alone is already two-thirds of 12, and \sqrt{80} is about 9, so the sum must be well over 12 ✓.
Tip:\sqrt{a} \cdot \sqrt{b} = \sqrt{ab} is a true rule. \sqrt{a} + \sqrt{b} = \sqrt{a + b} is false (unless one of them is 0). "Not" questions: evaluate every option and look for the odd one out.
Question 35 of 36
No calculatorFill in
Solve for the positive value of x.
3x^2 - 7 = 140
x =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Undo the steps in reverse order: undo the subtraction, then the multiplication by 3, and take the square root last.
Step-by-step solution
Undo "- 7": add 7 to both sides. 3x^2 = 147.
Undo "\times 3": divide both sides by 3. x^2 = 49.
Tip: The square root is always the last undo, because squaring happened first (before \times 3 and - 7). A common slip is to take \sqrt{147} or \sqrt{140} too early.
Question 36 of 36
No calculator
Two square garden plots are placed side by side, as shown. Their bottom edges are on the same line, and the whole right side of the smaller square touches the larger square.
What is the perimeter of the combined shape, in feet?
Hint
Find each side length from the area. Then trace around the outside of the shape, adding only the edges you walk along.
Step-by-step solution
Side lengths: small square \sqrt{49} = 7 ft; large square \sqrt{81} = 9 ft.
Trace the outline, starting at the bottom left and going clockwise:
left side of small square: 7
top of small square: 7
the step up the large square's left side above the small square: 9 - 7 = 2
top of large square: 9
right side of large square: 9
bottom of both squares: 7 + 9 = 16
Add: 7 + 7 + 2 + 9 + 9 + 16 = 50 feet.
Answer: 50 feet
Check: Another way: the two separate perimeters are 4 \times 7 = 28 and 4 \times 9 = 36, total 64. The touching edge (7 ft) is counted once on each square but is now inside, so subtract 2 \times 7 = 14: 64 - 14 = 50 ✓.
Tip: When shapes are joined, the shared edge disappears from the outside. Subtract it twice — once for each shape that counted it — and only the length that actually touches.
