Square Roots and Cube Roots
Squaring and taking a square root undo each other. So do cubing and taking a cube root. That is how you solve x² = p and x³ = p.
What you'll be able to do
- Find square roots and cube roots.
- Solve x² = p and x³ = p.
- Explain why x² = p can have two answers.
- Estimate a square root that is not a whole number.
- Find the sides of squares and the edges of cubes.
The big idea
Square roots. √49 = 7, because 7 × 7 = 49. The sign √ means the positive root only.
Two answers for x². x² = 49 has two answers, 7 and −7. Both make 49 when squared, because −7 × −7 = 49.
Cube roots. ∛64 = 4, because 4 × 4 × 4 = 64. A negative number has a cube root too. ∛(−8) = −2, because −2 × −2 × −2 = −8. So x³ = p has just one answer.
No answer. x² = −9 has no real answer. No number times itself makes a negative.
Do the other steps first. In 2x² + 3 = 75, take away 3. Then divide by 2. That gives x² = 36. Now take the root: x = 6 or x = −6.
Fractions and decimals. Take the root of the top and the bottom: √(4/9) = 2/3. Check decimals by squaring: 0.2 × 0.2 = 0.04, so √0.04 = 0.2.
Roots don't split over a sum. √(4 + 9) is √13. It is not 2 + 3 = 5.
Estimating. √30 is between √25 = 5 and √36 = 6. Try 5.5: 5.5 × 5.5 = 30.25. So √30 is about 5.5.
Irrational roots. Some roots, like √2, are irrational. Their decimals never end or repeat.
Words you need
- Square root (√)
- The positive number that, times itself, makes a number. √25 = 5.
- Cube root (∛)
- The number that, multiplied by itself three times, makes a number. ∛27 = 3.
- Perfect square
- A whole number times itself: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
- Perfect cube
- A whole number multiplied by itself three times: 1, 8, 27, 64, 125, 216.
- Irrational
- A number that can't be written as a fraction, like √2.
- Prime factors
- The primes that multiply to make a number. 12 = 2 × 2 × 3.
Two kinds of equations
| Equation | Answers | Why |
|---|---|---|
| x² = 49 | x = 7 or x = −7 | 7² and (−7)² are both 49 |
| x² = −49 | No real answer | No number squared is negative |
| x³ = 64 | x = 4 | Only 4³ is 64 |
| x³ = −64 | x = −4 | (−4)³ = −64 |
Worked examples
Example 1
Solve x² = 121.
- Which numbers squared make 121? 11 × 11 = 121.
- −11 × −11 = 121 too.
- So x = 11 or x = −11.
Answer: x = −11 or x = 11
Example 2
Solve x³ = −27.
- Which number cubed makes −27? 3 × 3 × 3 = 27.
- A negative cubed stays negative: (−3)³ = −27.
- So x = −3. It is the only answer.
Answer: x = −3
Example 3
A square tile has an area of 64 square inches. How long is each side?
- The area of a square is side × side.
- √64 = 8, because 8 × 8 = 64.
- A length can't be negative, so each side is 8 inches.
Answer: 8
Common mistakes
Giving one answer to x² = 36. Both 6 and −6 work. Give both, unless the question is about a length.
Dividing by 2: x² = 36 gives x = 18. The 2 means x times x. Undo it with a square root: √36 = 6.
Saying x³ = −8 has no answer. Negative numbers have cube roots. (−2)³ = −8, so x = −2.
Splitting a root over a sum: √(9 + 16) = 3 + 4. Add first. √(9 + 16) = √25 = 5.
Practice: 20 questions
6 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- Solve x² = 81.
- Solve x³ = −64.
- Solve x² = 25/49.
- A square rug has an area of 196 square feet. How long is each side, in feet? Word problem
- Kai says x² = 36 means x = 18. Find his mistake. What are the answers?
Tier 2 Intermediate One more step to think through
- Solve 3x² − 12 = 135.
- Solve 2x³ + 54 = 0.
- Which is greater, √(9 + 16) or √9 + √16?
- Solve x² = 0.09.
- A square tile has an area of 70 square inches. About how long is each side, to the nearest tenth of an inch? Word problem
Tier 3 Advanced Several steps - find the hidden structure
- A square garden has the same area as a rectangle 18 meters long and 8 meters wide. How long is each side of the square? Word problem
- A cube has a volume of 343 cubic centimeters. What is its total surface area? Word problem
- Find √(2⁸) + ∛(3⁶).
- A cube-shaped box has a total surface area of 150 square inches. What is its volume? Word problem
- Omar says √(16/81) = 16/9. Find his mistake. What is the correct value?
Tier 4 Challenge Stretch problems, some above grade level
- Put ∛340, √50 and 7.1 in order from least to greatest. Do it without a calculator.
- A cube's volume becomes 8 times as large. How many times as long does each edge become? Word problem
- Find the smallest whole number greater than 1 that is a perfect square. It must also be a perfect cube.
- A square has an area of 2 square feet. Can its side be written exactly as a fraction? Explain.
- Solve (2x − 3)² = 49. Above grade level
Answers
Tier 1 Foundations
- x = −9 or x = 9. 9 × 9 = 81, and (−9) × (−9) = 81 too. So there are two answers.
- x = −4. (−4) × (−4) × (−4) = −64. Only one number cubed makes −64.
- x = −5/7 or x = 5/7. 5/7 × 5/7 = 25/49, and (−5/7) × (−5/7) = 25/49 too. So there are two answers.
- 14. A root undoes squaring or cubing: √ undoes squaring, and ∛ undoes cubing. So √196 = 14.
- x = −6 or x = 6. Kai divided by 2. The 2 means x times x, so take the square root. 6 × 6 = 36, and (−6) × (−6) = 36 too. So there are two answers.
Tier 2 Intermediate
- x = −7 or x = 7. Add 12: 3x² = 147. Divide by 3: x² = 49. So x = 7 or x = −7. Check x = −7: left side 135, right side 135.
- x = −3. Take away 54: 2x³ = −54. Divide by 2: x³ = −27. So x = −3. Check x = −3: left side 0, right side 0.
- √9 + √16 is greater. √9 + √16 = 7, and √(9 + 16) = 5.
- x = −0.3 or x = 0.3. 0.3 × 0.3 = 0.09. So x = 0.3 or x = −0.3.
- about 8.4 inches. √70 is between √64 = 8 and √81 = 9. 8.4 × 8.4 = 70.56 and 8.3 × 8.3 = 68.89. So the side is about 8.4 inches.
Tier 3 Advanced
- 12. The rectangle's area is 18 × 8 = 144 square meters. √144 = 12, so each side is 12 meters.
- 294 square centimeters. Each edge is ∛343 = 7 cm, because 7 × 7 × 7 = 343. One face is 7 × 7 = 49. A cube has 6 faces, so 6 × 49 = 294.
- 25. √(2⁸) = 2⁴, because 4 + 4 = 8. ∛(3⁶) = 3², because 2 + 2 + 2 = 6. So 16 + 9 = 25.
- 125 cubic inches. A cube has 6 equal faces, so one face is 150 ÷ 6 = 25. Each edge is √25 = 5. The volume is 5 × 5 × 5 = 125.
- 4/9. Omar took the root of the bottom only. Take the root of both: √16 = 4 and √81 = 9. A root undoes squaring or cubing: √ undoes squaring, and ∛ undoes cubing. So √(16/81) = 4/9.
Tier 4 Challenge
- ∛340 < √50 < 7.1. 7 × 7 × 7 = 343, so ∛340 is just under 7. 7 × 7 = 49, so √50 is just over 7. And 7.1 × 7.1 = 50.41, so √50 is below 7.1.
- 2. The volume is edge × edge × edge. If each edge is 2 times as long, the volume grows 2 × 2 × 2 = 8 times.
- 64. Such a number is some number multiplied by itself six times. The smallest after 1 is 2⁶ = 64.
- No. Say a fraction a/b in lowest terms squared makes 2. Then a × a = 2 × b × b. Write both sides as prime factors. A square always has an even number of 2s. So the left side has an even number of 2s. The right side has an odd number. They can't be equal, so no fraction works. So the side, √2, is irrational.
- x = −2 or x = 5. Whatever is squared must be 7 or −7. So 2x − 3 = 7, giving x = 5. Or 2x − 3 = −7, giving x = −2. Check x = −2: left side 49, right side 49.
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