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Squares and Square Roots

Square numbers, find square roots of perfect squares, fractions and decimals, estimate roots that are not whole numbers, and avoid the classic sign traps.

Practice this skill 30 min Squares and square roots Leer en español
Key idea

What you'll learn

In this lesson you will learn to:

  • square any rational number — whole numbers, negatives, fractions and decimals;
  • find square roots of perfect squares, fractions and decimals;
  • estimate a square root that is not a whole number (for example, \sqrt{40} is between 6 and 7);
  • simplify square roots like \sqrt{72} = 6\sqrt{2};
  • solve simple equations like x^2 = 49.

Why it matters on the GED: squares and square roots show up everywhere — in area of squares, in the Pythagorean theorem, in the area of a circle (\pi r^2), in exponent questions and in the no-calculator section ("Which is closest to \sqrt{58}?"). The GED also likes to test the traps: -5^2 versus (-5)^2, and \sqrt{0.36} versus 0.06.

In real life: if you know a square room covers 144 square feet, the square root tells you each wall is 12 feet long — handy for buying baseboard, fencing or flooring.

Words to know

  • Square (of a number) — the number times itself: 7^2 = 7 \times 7 = 49.
  • Exponent — the small raised number that says how many times to use the base as a factor: in 5^2, the exponent is 2.
  • Base — the number being multiplied: in 5^2, the base is 5.
  • Perfect square — a number that is the square of an integer: 1, 4, 9, 16, 25, \dots (36 = 6^2).
  • Square root — a number that, when squared, gives the original number: a square root of 49 is 7, because 7^2 = 49.
  • Radical sign — the symbol \sqrt{\phantom{x}}. The number under it is the radicand: in \sqrt{30}, the radicand is 30.
  • Principal (positive) square root — the \sqrt{\phantom{x}} sign always means the non-negative root: \sqrt{49} = 7, not -7.
  • Irrational number — a number that cannot be written as a fraction of two integers; its decimal never ends or repeats: \sqrt{2} \approx 1.414\ldots
  • Simplified radical — a square root with no perfect-square factor left under the sign: \sqrt{72} = 6\sqrt{2}.

All Math words in the glossary

1. Squaring numbers

To square a number, multiply it by itself: n^2 = n \times n.

n1234567891011121314152025
n^2149162536496481100121144169196225400625

Memorize the squares from 1^2 to 15^2. They make roots and estimates fast.

A square with side length 5 units divided into a 5 by 5 grid of 25 unit squares, showing that 5 squared equals 25.5 units5 unitsArea = 5² = 25 square units

The picture shows why it is called "squaring": a square with side 5 is made of 5^2 = 25 unit squares. Area of a square = s^2.

Squaring is not doubling. 6^2 = 36, not 12.

Negatives. A negative times a negative is positive, so the square of any non-zero number is positive: (-6)^2 = (-6)(-6) = 36.

Watch the parentheses! The exponent only applies to what is directly next to it.

  • (-6)^2 = 36 (the base is -6)
  • -6^2 = -(6^2) = -36 (the base is 6; the minus sign is applied afterwards)

Fractions. Square the top and the bottom: \left(\tfrac{2}{3}\right)^2 = \tfrac{4}{9}.

A unit square cut into a 3 by 3 grid of 9 equal parts. A 2 by 2 block in the corner is shaded blue; its sides are each 2/3 of the whole side. 4 of the 9 parts are shaded, showing (2/3)² = 4/9.(2/3)² = 2/3 × 2/3 = 4/92/32/3whole square: side 1, 9 equal parts4 of the 9 parts are shaded: 4/9

Decimals. Square the digits, then double the number of decimal places: 0.3^2 = 0.09 (not 0.9), 1.2^2 = 1.44, 0.05^2 = 0.0025.

Zero and one: 0^2 = 0 and 1^2 = 1. A number between 0 and 1 gets smaller when squared: 0.5^2 = 0.25.

2. Square roots

The square root "undoes" squaring. \sqrt{81} asks: what non-negative number times itself is 81? Answer: 9.

  • \sqrt{0} = 0, \sqrt{1} = 1, \sqrt{144} = 12, \sqrt{400} = 20.

Fractions: take the root of the top and the bottom.

\sqrt{\frac{49}{64}} = \frac{\sqrt{49}}{\sqrt{64}} = \frac{7}{8}

Decimals: the root has half as many decimal places. Write it as a fraction if unsure:

\sqrt{0.36} = \sqrt{\tfrac{36}{100}} = \tfrac{6}{10} = 0.6 \qquad \sqrt{0.0049} = 0.07

(Check: 0.6^2 = 0.36 ✓. A common wrong answer is 0.06, but 0.06^2 = 0.0036.)

Negatives under the root. No real number squared is negative, so \sqrt{-49} is not a real number. But -\sqrt{49} = -7 is fine (the minus is outside).

Equations. x^2 = 49 has two solutions: x = 7 and x = -7, because both square to 49. We write x = \pm 7. The symbol \sqrt{49} alone means only +7.

Number line from −10 to 10 with dots at −7 (red) and 7 (blue), each 7 units from zero. Labels show (−7)² = 49 and 7² = 49, so x² = 49 has the two solutions 7 and −7, while √49 means only 7.x² = 49 has two solutions: x = 7 or x = −7−10−9−8−7−6−5−4−3−2−1012345678910(−7)² = 497² = 49But the symbol √49 means only the positive root: √49 = 7

Products and quotients split; sums do not.

  • \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}: \sqrt{4 \cdot 25} = 2 \cdot 5 = 10 ✓
  • \sqrt{a + b} \ne \sqrt{a} + \sqrt{b}: \sqrt{16 + 9} = \sqrt{25} = 5, but \sqrt{16} + \sqrt{9} = 7 ✗

3. Estimating and simplifying square roots

Estimating. Most numbers are not perfect squares. To estimate \sqrt{40}:

  1. Find the perfect squares on either side: 36 < 40 < 49.
  2. Take roots: 6 < \sqrt{40} < 7.
  3. 40 is closer to 36 (4 away) than to 49 (9 away), so \sqrt{40} is closer to 6. In fact \sqrt{40} \approx 6.32.
Number line from 0 to 10. The square root of 40, about 6.32, is marked between 6 and 7, closer to 6, because 40 lies between the perfect squares 36 and 49.012345678910√40 ≈ 6.32(√36)(√49)

Squares get further apart as numbers grow, so the root is not exactly in proportion — the estimate is a rough guide, and the calculator gives the exact decimal.

Simplifying. Look for the largest perfect-square factor of the radicand and pull its root out front:

\sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2} \sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2} \qquad \sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3}

If you use a smaller perfect square (\sqrt{72} = \sqrt{4 \cdot 18} = 2\sqrt{18}), the answer is still equal but not fully simplified; keep going (2\sqrt{18} = 2 \cdot 3\sqrt{2} = 6\sqrt{2}).

Working backwards from area. If a square has area A, its side is s = \sqrt{A}. A square patio of 225\text{ ft}^2 has sides of \sqrt{225} = 15\text{ ft}.

Formula

Formulas and rules

a^2 = a \cdot a \qquad \sqrt{a^2} = a \text{ (for } a \ge 0\text{)} \qquad \left(\sqrt{a}\right)^2 = a \sqrt{ab} = \sqrt{a}\,\sqrt{b} \qquad \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \quad (a \ge 0,\ b > 0) x^2 = k \;(k > 0) \;\Rightarrow\; x = \sqrt{k} \text{ or } x = -\sqrt{k}

Area of a square: A = s^2, so s = \sqrt{A} (s = side length, A = area).

Formula sheet: the area of a square (A = s^2) is on the GED formula sheet, and so is the Pythagorean theorem, which uses squares and roots. The rules for working with squares and roots above are not on the sheet — memorize them, along with the perfect squares up to 15^2 = 225.

Worked example

Example 1 (easy): the sign trap

Evaluate (-7)^2 - 3^2.

A 7 by 7 grid of 49 unit squares shows (−7)² = 49. A 3 by 3 block of 9 squares in the top-right corner is outlined in red and crossed out, showing 3² = 9 removed. The 40 blue squares left show (−7)² − 3² = 49 − 9 = 40.(−7)² − 3² = 49 − 9 = 407733remove3² = 97 × 7 square = 49 unit squares; 49 − 9 = 40 left (blue)
  1. (-7)^2 = (-7)(-7) = 49 (the base is -7; negative × negative = positive).
  2. 3^2 = 3 \times 3 = 9.
  3. 49 - 9 = 40.

Answer: 40

Check: On the calculator, type ( (−) 7 ) x² − 3 x² → 40 ✓.

Worked example

Example 2 (word problem): side of a square

A square garden has an area of 144 square feet. How much fencing is needed to go all the way around it?

A square garden with area 144 square feet. Each side is labelled 12 feet and each corner has a right-angle mark. A dashed blue line around the outside shows the fence: perimeter = 4 × 12 = 48 feet.Square garden: A = s² = 144 ft²Area = 144 ft²side = √144 = 12 ft12 ft12 ft12 ft12 ftFence (blue dashes) = 4 × 12 = 48 ft
  1. Area of a square: A = s^2, so s^2 = 144.
  2. Side: s = \sqrt{144} = 12 feet (a length cannot be negative, so use only +12).
  3. Perimeter: 4s = 4 \times 12 = 48 feet.

Answer: 48 feet of fencing

Check: 12 \times 12 = 144 ✓, and four sides of 12 ft make 48 ft ✓.

Worked example

Example 3: roots of fractions and decimals

Evaluate \sqrt{\tfrac{25}{81}} + \sqrt{0.09}. Give the answer as a fraction.

Two unit squares. Left: a 9 by 9 grid with a 5 by 5 corner shaded blue, 25 of 81 small squares, so a square of area 25/81 has side 5/9. Right: a 10 by 10 grid with a 3 by 3 corner shaded green, 9 of 100 small squares, so a square of area 0.09 has side 0.3. The sum is 5/9 + 3/10 = 77/90.√(25/81) = 5/9 and √0.09 = √(9/100) = 3/10 = 0.3side 5/925 of 81 squares = 25/81side 3/10 = 0.39 of 100 squares = 0.09Sum: 5/9 + 3/10 = 50/90 + 27/90 = 77/90
  1. \sqrt{\tfrac{25}{81}} = \tfrac{\sqrt{25}}{\sqrt{81}} = \tfrac{5}{9}.
  2. \sqrt{0.09} = \sqrt{\tfrac{9}{100}} = \tfrac{3}{10} (that is, 0.3).
  3. Add with LCD 90: \tfrac{5}{9} + \tfrac{3}{10} = \tfrac{50}{90} + \tfrac{27}{90} = \tfrac{77}{90}.

Answer: \tfrac{77}{90}

Check: \tfrac{5}{9} \approx 0.556 and 0.556 + 0.3 = 0.856; 77 \div 90 \approx 0.856 ✓.

Worked example

Example 4 (GED-hard): estimate without a calculator

Between which two consecutive whole numbers is \sqrt{90}, and which one is it closer to?

Two linked number lines. The top line shows squares from 81 to 100 with 90 marked: 90 is 9 away from 81 and 10 away from 100. The bottom line shows numbers from 9 to 10 with 9.5 marked; an arrow from 90 points to √90 ≈ 9.49, just left of 9.5 and a little closer to 9.81 < 90 < 100, so 9 < √90 < 10x²x819010099.510√90 ≈ 9.499 away10 away90 is a little closer to 81, so √90 is a little closer to 9 (9.5² = 90.25)
  1. Perfect squares near 90: 9^2 = 81 and 10^2 = 100.
  2. 81 < 90 < 100, so 9 < \sqrt{90} < 10.
  3. Distance: 90 - 81 = 9 and 100 - 90 = 10. 90 is slightly closer to 81, so \sqrt{90} is slightly closer to 9.

Answer: between 9 and 10, a little closer to 9 (calculator: \sqrt{90} \approx 9.49).

Check: 9.5^2 = 90.25, which is just above 90, so \sqrt{90} is just under 9.5 ✓.

Worked example

Example 5 (work backwards, word problem): from cost to size

Tile for a square floor costs $4.50 per square foot. The tile for the whole floor cost $288. How long is each side of the floor?

A square floor drawn as an 8 by 8 grid of 64 one-square-foot tiles. Each side is labelled 8 feet. 288 dollars ÷ 4.50 dollars = 64 square feet, and √64 = 8 feet.Tile cost $288 ÷ $4.50 per ft² = 64 ft²8 ft8 ft8 × 8 = 64 tiles of 1 ft², so s = √64 = 8 ft
  1. Find the area: 288 \div 4.50 = 64 square feet.
  2. Side of a square: s = \sqrt{64} = 8 feet.

Answer: 8 feet

Check: 8 \times 8 = 64 sq ft, and 64 \times \$4.50 = \$288 ✓.

Calculator tip

Using the TI-30XS MultiView

  • Square: type the number, then press x². For a negative, use parentheses and the (−) key: ( (−) 5 ) x² gives 25. Without parentheses, (−) 5 x² gives -25 — exactly the -5^2 trap.
  • Square root: press 2nd then x² to get \sqrt{\phantom{x}}, type the number, press the right arrow to close the root, then enter. Example: 2nd x² 40 enter. In MathPrint mode the answer may appear as the simplified root 2\sqrt{10}; press toggle to see the decimal 6.32455532.
  • Fractions under a root: press 2nd x², then n/d, enter 49 and 64, then enter; the result shows \tfrac{7}{8}. Press toggle to see 0.875.
  • Simplified radicals: the calculator may show an answer like 6\sqrt{2} in MathPrint mode; press toggle to see the decimal 8.485\ldots and compare with the answer choices.

When NOT to use it: questions about perfect squares (\sqrt{144}), estimates ("between which two integers") and sign rules often appear in the no-calculator section. Know your squares to 15^2 so these take seconds.

Common mistake

Common mistakes

  • Doubling instead of squaring. Wrong: 6^2 = 12. Right: 6^2 = 6 \times 6 = 36.
  • Halving instead of rooting. Wrong: \sqrt{64} = 32. Right: \sqrt{64} = 8 because 8 \times 8 = 64.
  • The negative-base trap. Wrong: -5^2 = 25. Right: -5^2 = -25, while (-5)^2 = 25.
  • Decimal places. Wrong: \sqrt{0.36} = 0.06 and 0.3^2 = 0.9. Right: \sqrt{0.36} = 0.6 and 0.3^2 = 0.09 (square → double the decimal places; root → half).
  • Splitting a sum. Wrong: \sqrt{16 + 9} = 4 + 3 = 7. Right: add first, \sqrt{25} = 5.
  • Forgetting the negative solution — or inventing one. x^2 = 49 has two answers, \pm 7; but \sqrt{49} by itself is only 7. And \sqrt{-49} is not a real number (it is not -7).
Test strategy

GED test strategy

  • Use the perfect-square "fence". For any root, write the perfect squares just below and above. This eliminates most wrong choices in seconds, without a calculator.
  • Square the answer choices. For "what is \sqrt{\ldots}" questions, squaring each choice and comparing is often fastest — and it catches decimal-place errors.
  • Check reasonableness. A side length found from an area must be much smaller than the area (for areas bigger than 1). If you got \sqrt{144} = 72, something is off.
  • Look for parentheses around negative numbers before squaring. Read -x^2 and (-x)^2 slowly.
  • Read what is asked. Area problems may ask for the side, the perimeter or a cost — finding the root is often only step one.
  • Units: area is in square units (\text{ft}^2); the root of an area is in plain units (ft).

Check yourself

Squares and square rootsNo calculator

What is the value of \sqrt{81}?

Answer choices
Hint

Which number, multiplied by itself, gives 81?

Check yourself

Squares and square rootsNo 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?

Summary

Recap

  • n^2 = n \times n; memorize squares up to 15^2 = 225. Squaring is not doubling.
  • \sqrt{\phantom{x}} means the non-negative root: \sqrt{81} = 9. But x^2 = 81 has two solutions, x = \pm 9.
  • (-5)^2 = 25 but -5^2 = -25; \sqrt{-25} is not a real number.
  • Fractions: root (or square) top and bottom. Decimals: squaring doubles decimal places, rooting halves them.
  • Estimate a root between the two nearest perfect squares; simplify by pulling out the largest perfect-square factor.
  • \sqrt{ab} = \sqrt{a}\sqrt{b}, but \sqrt{a + b} \ne \sqrt{a} + \sqrt{b}.
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