Topic Quiz Q.2: Operations with rational numbers, squares, cubes and roots
No calculator in this part
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Question 1 of 10 · When is an expression undefined?
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Look at each fraction below. Choose all of the ones that have a zero denominator and so have no value.
Select all that apply.
Hint
Look only at each denominator after simplifying it. Zero on top is allowed.
Step-by-step solution
Simplify each denominator and look for 0.
\dfrac{12}{0}: denominator 0 → undefined.
\dfrac{0}{12} = 0 → defined (zero on top is fine, since 0 \times 12 = 0).
\dfrac{0}{0}: denominator 0 → undefined (the GED treats every zero denominator as undefined).
\dfrac{12}{4 - 4} = \dfrac{12}{0} → undefined.
\dfrac{4 - 4}{12} = \dfrac{0}{12} = 0 → defined.
\dfrac{-12}{-12} = 1 → defined.
Answer:\dfrac{12}{0}, \dfrac{0}{0} and \dfrac{12}{4 - 4}.
Check: Each chosen expression has a denominator equal to 0; the three others have values 0, 0 and 1 ✓.
Tip: Read every fraction "bottom first". If the bottom is 0 (even after a little arithmetic), it is undefined — no matter what is on top.
Question 2 of 10 · Add, subtract, multiply and divide rational numbers
At 6 a.m. the temperature in Fargo was 7^\circ\text{F}. By noon it had dropped 15^\circ\text{F}, and by 6 p.m. it had risen 9^\circ\text{F} from the noon temperature. What was the temperature at 6 p.m.?
Hint
A drop is a negative change; a rise is a positive change. Start at 7 and apply each change in order.
Step-by-step solution
Start at 7^\circF.
A drop of 15^\circ means subtract: 7 - 15 = -8. The noon temperature was -8^\circF (below zero).
A rise of 9^\circ means add: -8 + 9 = 1. Different signs: 9 - 8 = 1, keep the sign of 9 (positive).
Answer: 1^\circ\text{F}
Check: The overall change is -15 + 9 = -6, and 7 - 6 = 1 ✓.
Tip: Turn words into signs: "dropped", "fell", "lost", "withdrew", "owed" → negative; "rose", "gained", "deposited", "earned" → positive. Then just add the signed numbers.
Question 3 of 10 · Add, subtract, multiply and divide rational numbers
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Select all of the expressions that have a negative value.
Select all that apply.
Hint
Work out the sign (and the value) of each expression. Count negative factors in products: an even number of negatives gives a positive.
Check: For each sum, compare sizes: \tfrac{7}{8} > \tfrac{3}{4} and the larger one is negative, so the sum is negative ✓; 1\tfrac{1}{6} > \tfrac{2}{3}, so subtracting it gives a negative ✓.
Tip: You often only need the sign, not the exact value. For products count the negatives; for sums ask which part is bigger.
Question 4 of 10 · When is an expression undefined?
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Three friends are hosting a party and will not pay. The other guests split the $360 cost equally. If n is the total number of people at the party (including the 3 hosts), the cost per paying guest, in dollars, is
C = \frac{360}{n - 3}
For what value of n is this formula undefined?
Hint
Which value of n makes the denominator zero? What would that mean at the party?
Step-by-step solution
The formula is undefined when the denominator is 0: n - 3 = 0.
Solve: n = 3.
Meaning: if only 3 people come, they are all hosts, so nobody is paying — there is no "cost per paying guest".
Answer: n = 3
Check:C = \dfrac{360}{3 - 3} = \dfrac{360}{0} → undefined ✓. With n = 12, C = \tfrac{360}{9} = 40, a sensible $40 each.
Tip: In a real-world formula, the "undefined" value usually describes a situation that cannot happen or makes no sense (no paying guests, zero time, zero people). Setting the denominator to 0 finds it.
Question 5 of 10 · Multi-step real-world problems
Fill in
Ana is driving 378 miles to visit family. Her car gets 28 miles per gallon, and gas costs $3.60 per gallon. How much will the gas for the trip cost?
$
Type your answer in the box. Fractions like 3/4 are fine.
Hint
First find how many gallons she needs. Then find what those gallons cost.
Step-by-step solution
Gallons needed = miles ÷ miles per gallon: 378 \div 28 = 13.5 gallons.
Check: Work backwards: \$48.60 \div \$3.60 = 13.5 gallons, and 13.5 \times 28 = 378 miles ✓.
Tip: Follow the units. \text{miles} \div \tfrac{\text{miles}}{\text{gallon}} = \text{gallons}, and \text{gallons} \times \tfrac{\$}{\text{gallon}} = \$. If the units don't come out to what the question asks for, the operation is wrong (for example, 378 \times 3.60 gives "mile-dollars per gallon", which makes no sense).
Question 6 of 10 · Multi-step real-world problems
Drop-down
Select the correct answer from each drop-down menu.
Five friends go camping. The campsite costs $42 per night for 3 nights. They also spend $97.50 on groceries and $18 on firewood. They split all costs equally.
The total cost of the trip is .
Each friend pays .
Choose an answer from each menu.
Hint
Find the campsite cost for all 3 nights first, then add the other costs. Divide by the number of people, not the number of nights.
Step-by-step solution
Campsite: 42 \times 3 = 126.
Total: 126 + 97.50 + 18 = 241.50.
Each friend: 241.50 \div 5 = 48.30.
Answer: total $241.50; each friend pays $48.30.
Check:5 \times 48.30 = 241.50 ✓.
Tip: Watch for the two different counts in this problem: 3 nights (multiply the nightly price) and 5 friends (divide the total). Mixing them up is the most common mistake.
Question 7 of 10 · Squares and square roots
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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 8 of 10 · Cubes and cube roots
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A candle company sells two cube-shaped candles. The small candle has a volume of 8 cubic inches, and the large candle has a volume of 216 cubic inches. How many times as long is an edge of the large candle as an edge of the small candle?
Hint
Find each edge with a cube root, then compare the edges by dividing.
Step-by-step solution
Small edge:\sqrt[3]{8} = 2 inches.
Large edge:\sqrt[3]{216} = 6 inches (because 6^3 = 216).
Compare:6 \div 2 = 3. The large edge is 3 times as long.
Answer: 3 times as long
Check: The volume ratio is 216 \div 8 = 27, and \sqrt[3]{27} = 3 ✓. Edges scaled by 3 make the volume 3^3 = 27 times as large.
Tip: Volume ratio = (edge ratio)^3, so edge ratio = \sqrt[3]{\text{volume ratio}}. "How many times as long" asks for a ratio (divide), not a difference (subtract).
Question 9 of 10 · Squares and square roots
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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 10 of 10 · Cubes and cube roots
No calculatorDrop-down
Select the correct answer from each drop-down menu.
If x^3 = -0.064, then x = . If 4y^3 = 500, then y = .
Choose an answer from each menu.
Hint
For each equation, get the cube by itself, then take the cube root. Keep the sign, and count decimal places.
Tip: For a decimal like 0.064, ignore the decimal point, take the cube root of 64 (which is 4), then give the answer one-third as many decimal places: 3 places → 1 place, so 0.4.