Square Roots and Cube Roots

Learn how to find square roots and cube roots, the inverse operations of squaring and cubing numbers.

Intermediate25 minLesson
Printable worksheet

Definition

A square root of a number is a value that, when multiplied by itself, gives the original number. We write it with the radical symbol: n\sqrt{n}
25=5 because 5×5=25\sqrt{25} = 5 \text{ because } 5 \times 5 = 25
A cube root is a value that, used as a factor three times, gives the original number. We write it as: n3\sqrt[3]{n}
83=2 because 2×2×2=8\sqrt[3]{8} = 2 \text{ because } 2 \times 2 \times 2 = 8
Key Insight: Square roots and cube roots are the inverse operations of squaring and cubing:
  • If 52=255^2 = 25, then 25=5\sqrt{25} = 5
  • If 23=82^3 = 8, then 83=2\sqrt[3]{8} = 2

Try it now

What is 36\sqrt{36}?

Worked Examples

What is 81\sqrt{81}?
1

Understand the question

We need a number that, multiplied by itself, gives 81 → Find nn where n×n=81n \times n = 81
2

Think of perfect squares

1,4,9,16,25,36,49,64,81,...1, 4, 9, 16, 25, 36, 49, 64, 81, ... → 81 is in this list!
3

Identify the root

9×9=819 \times 9 = 81 → 81=9\sqrt{81} = 9

Common Mistakes

Thinking 25+144=25+144=5+12=17\sqrt{25 + 144} = \sqrt{25} + \sqrt{144} = 5 + 12 = 17

Why it's wrong: You cannot split a square root across addition! 25+144=169=13\sqrt{25 + 144} = \sqrt{169} = 13, not 17.

Correct: Calculate inside the radical first, then take the root: 169=13\sqrt{169} = 13

Confusing square and cube roots

Why it's wrong: Square root asks 'what times itself equals this?' Cube root asks 'what times itself times itself equals this?'

Correct: 8≈2.83\sqrt{8} \approx 2.83 (not 2), but 83=2\sqrt[3]{8} = 2 exactly

Forgetting that x2=∣x∣\sqrt{x^2} = |x|, not just xx

Why it's wrong: Both 525^2 and (−5)2(-5)^2 equal 25, but 25\sqrt{25} gives the positive answer.

Correct: The principal square root is always non-negative: 25=5\sqrt{25} = 5, not ±5\pm 5

Interactive Visual

3² = 9
3

See how powers of a number grow on the number line. Change the base and exponent.

3³ = 27
3

See how powers of a number grow on the number line. Change the base and exponent.

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Practice Problems

16 problems
Problem 1 of 16
Easy
What is 36\sqrt{36}?

Why It Matters

Square roots and cube roots appear constantly in mathematics and everyday life:
  • Geometry: Finding the side length of a square when you know its area (A=s2A = s^2, so s=As = \sqrt{A})
  • Volume problems: Finding the edge of a cube when you know its volume (V=s3V = s^3, so s=V3s = \sqrt[3]{V})
  • Physics: Calculating distance, speed, and many natural phenomena
  • Construction: Architects and builders use square roots for measurements and the Pythagorean theorem
Understanding roots unlocks the ability to solve equations like x2=49x^2 = 49 and x3=125x^3 = 125!

Real World Applications

Architecture and Construction

Architects use square roots to calculate diagonal measurements and verify right angles using the Pythagorean theorem.

Example:

To check if a corner is square, builders measure 3 meters on one side, 4 meters on the other. The diagonal should be 32+42=25=5\sqrt{3^2 + 4^2} = \sqrt{25} = 5 meters.

1Try It Yourself

A rectangular room is 6 meters by 8 meters. You want to put a diagonal support beam.

How long should the beam be?

Step 1: Write the mathematical expression

Use the Pythagorean theorem: 62+82\sqrt{6^2 + 8^2}

Shipping and Packaging

Companies calculate box dimensions from volume requirements using cube roots.

Example:

If you need a cubic box with 1000 cubic centimeters of space, each edge should be 10003=10\sqrt[3]{1000} = 10 cm.

2Try It Yourself

A company ships products in cubic boxes. They need boxes with exactly 343 cubic centimeters of space.

What should each edge of the box measure?

Step 1: Write the mathematical expression

Find 3433\sqrt[3]{343}

Key Takeaways

  • 1The square root n\sqrt{n} is the number that, when squared, gives nn
  • 2The cube root n3\sqrt[3]{n} is the number that, when cubed, gives nn
  • 3Perfect squares: 1,4,9,16,25,36,49,64,81,100,...1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...
  • 4Perfect cubes: 1,8,27,64,125,216,343,512,729,1000,...1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, ...
  • 5To estimate non-perfect roots, find the two perfect squares or cubes it falls between

Frequently Asked Questions

They mean the same thing! x=x1/2\sqrt{x} = x^{1/2}. Similarly, x3=x1/3\sqrt[3]{x} = x^{1/3}. This is useful in algebra.
They mean the same thing! x=x1/2\sqrt{x} = x^{1/2}. Similarly, x3=x1/3\sqrt[3]{x} = x^{1/3}. This is useful in algebra.
Not with real numbers. There's no real number that multiplied by itself gives a negative result. (In advanced math, 'imaginary numbers' handle this.)
2≈1.414...\sqrt{2} \approx 1.414... never ends or repeats. It cannot be written as a simple fraction, making it an irrational number.

Glossary

Square root
A number that when multiplied by itself gives the original number; written as n\sqrt{n}
Cube root
A number that, used as a factor three times, gives the original number; written as n3\sqrt[3]{n}
Perfect square
A number that is the square of a whole number (e.g., 1, 4, 9, 16, 25)
Perfect cube
A number that is the cube of a whole number (e.g., 1, 8, 27, 64, 125)
Radical
The symbol \sqrt{} used to denote roots
Principal root
The non-negative square root of a number

Formula Card

Square Root

n=x\sqrt{n} = x means x2=nx^2 = n

$x$ times itself equals $n$

Cube Root

n3=x\sqrt[3]{n} = x means x3=nx^3 = n

$x$ times itself times itself equals $n$

Exponent Form

n=n1/2\sqrt{n} = n^{1/2}, n3=n1/3\sqrt[3]{n} = n^{1/3}

Roots as fractional exponents

Root of a Product

a×b=a×b\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}

Split a product under the radical (for $a, b \ge 0$)

Today's challengeOne problem, three tries

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