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:
A cube root is a value that, when multiplied by itself three times, gives the original number. We write it as:
Key Insight: Square roots and cube roots are the inverse operations of squaring and cubing:
  • If , then
  • If , then

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What is ?

Worked Examples

What is ?

1

Understand the question

We need a number that, multiplied by itself, gives 81 → Find where

2

Think of perfect squares

→ 81 is in this list!

3

Identify the root

→

Common Mistakes

Thinking

Why it's wrong: You cannot split a square root across addition! , not 17.

Correct: Calculate inside the radical first, then take the root:

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: (not 2), but exactly

Forgetting that , not just

Why it's wrong: Both and equal 25, but gives the positive answer.

Correct: The principal square root is always non-negative: , not

Interactive Visual

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Practice Problems

16 problems
Problem 1 of 16
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What is ?

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 (, so )
  • Volume problems: Finding the edge of a cube when you know its volume (, so )
  • 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 and !

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 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:

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 cm.

2Try It Yourself

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

What should each edge of the box measure?

Step 1: Write the mathematical expression

Find

Key Takeaways

  • 1The square root is the number that, when squared, gives
  • 2The cube root is the number that, when cubed, gives
  • 3Perfect squares:
  • 4Perfect cubes:
  • 5To estimate non-perfect roots, find the two perfect squares or cubes it falls between

Frequently Asked Questions

They mean the same thing! . Similarly, . This is useful in algebra.
They mean the same thing! . Similarly, . 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.)
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
Cube root
A number that when multiplied by itself three times gives the original number; written as
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 used to denote roots
Principal root
The non-negative square root of a number

Formula Card

Square Root

means

$x$ times itself equals $n$

Cube Root

means

$x$ times itself times itself equals $n$

Exponent Form

,

Roots as fractional exponents

Product Rule

Split products inside radicals

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