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Square & Cube
Roots

Master the inverse of squaring and cubing: learn to evaluate perfect squares, perfect cubes, and discover why \(\sqrt{2}\) is irrational.

8th Grade Math

What Is a Square Root?

A square root of a number \(p\) is the value that, when multiplied by itself, gives \(p\). We write it as \(\sqrt{p}\).

25
side = 5
\(x^2 = 25\)
\(x = \sqrt{25} = 5\)

What Is a Cube Root?

A cube root of a number \(p\) is the value that, when used as a factor three times, gives \(p\). We write it as \(\sqrt[3]{p}\).

27
edge = 3
\(x^3 = 27\)
\(x = \sqrt[3]{27} = 3\)

Why Do Roots Matter?

Roots are the inverse operations of squaring and cubing. Whenever you need to undo an exponent, you reach for a root.

Geometry
Find the side of a square given its area, or the edge of a cube given its volume.
Science
Calculate distance, speed, or force when formulas involve squared or cubed terms.
Algebra
Solve equations like \(x^2 = p\) and \(x^3 = p\) for an unknown variable.

The Perfect Squares

A perfect square is a number that equals some integer times itself. You should memorize these small perfect squares:

\(1^2 = 1\)
\(\sqrt{1} = 1\)
\(2^2 = 4\)
\(\sqrt{4} = 2\)
\(3^2 = 9\)
\(\sqrt{9} = 3\)
\(4^2 = 16\)
\(\sqrt{16} = 4\)
\(5^2 = 25\)
\(\sqrt{25} = 5\)
\(6^2 = 36\)
\(\sqrt{36} = 6\)
\(7^2 = 49\)
\(\sqrt{49} = 7\)
\(8^2 = 64\)
\(\sqrt{64} = 8\)
\(9^2 = 81\)
\(\sqrt{81} = 9\)
\(10^2=100\)
\(\sqrt{100}=10\)
\(11^2=121\)
\(\sqrt{121}=11\)
\(12^2=144\)
\(\sqrt{144}=12\)

The Perfect Cubes

A perfect cube is a number that equals some integer multiplied by itself three times. Memorize these:

\(1^3 = 1\)
\(\sqrt[3]{1} = 1\)
\(2^3 = 8\)
\(\sqrt[3]{8} = 2\)
\(3^3 = 27\)
\(\sqrt[3]{27} = 3\)
\(4^3 = 64\)
\(\sqrt[3]{64} = 4\)
\(5^3=125\)
\(\sqrt[3]{125}=5\)
\(6^3=216\)
\(\sqrt[3]{216}=6\)
\(7^3=343\)
\(\sqrt[3]{343}=7\)
\(8^3=512\)
\(\sqrt[3]{512}=8\)
\(10^3=1000\)
\(\sqrt[3]{1000}=10\)

The Core Equations

The two fundamental equation types you must solve:

Square Root
If \(x^2 = p\), then \(x = \sqrt{p}\).
Both \(+\sqrt{p}\) and \(-\sqrt{p}\) are solutions, but \(\sqrt{p}\) means the positive root.
Cube Root
If \(x^3 = p\), then \(x = \sqrt[3]{p}\).
Cube roots have exactly one real solution (positive when \(p > 0\)).

Square vs Cube Roots

Square Root \(\sqrt{\ }\) Cube Root \(\sqrt[3]{\ }\)
Solves \(x^2 = p\) \(x^3 = p\)
Number of real solutions Two (\(\pm\sqrt{p}\)) One
Input restriction \(p \geq 0\) Any real \(p\)
Geometric meaning Side of a square Edge of a cube
Example \(\sqrt{49} = 7\) \(\sqrt[3]{125} = 5\)

Rational vs Irrational Roots

The square root of a perfect square is a rational number (it can be written as a fraction). But when \(p\) is not a perfect square, \(\sqrt{p}\) is irrational: its decimal goes on forever without repeating.

Rational
\(\sqrt{16} = 4\)
\(\sqrt{0.25} = 0.5\)
\(\sqrt{\tfrac{9}{4}} = \tfrac{3}{2}\)
Irrational
\(\sqrt{2} \approx 1.41421...\)
\(\sqrt{3} \approx 1.73205...\)
\(\sqrt{5} \approx 2.23606...\)

How to Evaluate Roots

1
Identify the type: Is it a square root \(\sqrt{\ }\) or a cube root \(\sqrt[3]{\ }\)?
2
Check for a perfect power: Is the number under the radical a perfect square (or perfect cube)?
3
Find the root: Ask "what number times itself (twice or three times) gives this value?"
4
Consider solutions: For \(x^2 = p\), remember there are two solutions: \(\pm\sqrt{p}\). For \(x^3 = p\), there is one.
5
Not a perfect power? State whether the root is rational or irrational. Approximate if asked.

Example 1: Evaluating \(\sqrt{81}\)

81
? units
Step 1: \(x^2 = 81\)
Step 2: Is 81 a perfect square?
\(9 \times 9 = 81\) ✓
Step 3: \(\sqrt{81} = 9\)
Solutions to \(x^2 = 81\): \(x = +9\) or \(x = -9\)

Example 2: Evaluating \(\sqrt[3]{64}\)

64
Step 1: \(x^3 = 64\)
Step 2: Is 64 a perfect cube?
\(4 \times 4 \times 4 = 64\) ✓
Step 3: \(\sqrt[3]{64} = 4\)
Only one solution: \(x = 4\)

Example 3: Fraction Under the Radical

Evaluate \(\sqrt{\dfrac{25}{49}}\).

1
Separate the root: \(\sqrt{\dfrac{25}{49}} = \dfrac{\sqrt{25}}{\sqrt{49}}\)
2
Evaluate each part: \(\sqrt{25} = 5\) and \(\sqrt{49} = 7\)
3
Combine: \(\sqrt{\dfrac{25}{49}} = \dfrac{5}{7}\)

Example 4: Is \(\sqrt{10}\) Rational or Irrational?

We need to determine whether \(\sqrt{10}\) can be expressed as a fraction of two integers.

1
Check perfect squares near 10: \(3^2 = 9\) and \(4^2 = 16\). Since \(10\) is between them, it is not a perfect square.
2
Conclude: \(\sqrt{10}\) is irrational. Its decimal expansion never terminates or repeats.
3
Approximate: \(3 < \sqrt{10} < 4\). A calculator gives \(\sqrt{10} \approx 3.16227...\)

Common Mistakes

✕ Forgetting the negative root
\(x^2 = 49\) has two solutions: \(x = 7\) and \(x = -7\). Many students only write \(x = 7\).
✕ Halving instead of rooting
Thinking \(\sqrt{36} = 18\) (dividing by 2). A square root asks "what times itself?" not "divide by 2."
✕ Confusing \(\sqrt{\ }\) and \(\sqrt[3]{\ }\)
\(\sqrt{64} = 8\) but \(\sqrt[3]{64} = 4\). The index matters: squaring vs cubing are different operations.

Quick Reference

Concept Rule / Fact Example
Square root \(\sqrt{p}\) : \(x \times x = p\) \(\sqrt{144} = 12\)
Cube root \(\sqrt[3]{p}\) : \(x \times x \times x = p\) \(\sqrt[3]{125} = 5\)
Two solutions \(x^2 = p \Rightarrow x = \pm\sqrt{p}\) \(x^2 = 25 \Rightarrow x = \pm 5\)
One solution \(x^3 = p \Rightarrow x = \sqrt[3]{p}\) \(x^3 = 8 \Rightarrow x = 2\)
Fraction rule \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\) \(\sqrt{\frac{9}{16}} = \frac{3}{4}\)
Irrational roots \(\sqrt{p}\) is irrational if \(p\) is not a perfect square \(\sqrt{2} \approx 1.414...\)

Question 1 of 5

What is \(\sqrt{121}\)?

Question 2 of 5

What is \(\sqrt[3]{216}\)?

Question 3 of 5

What are all solutions to the equation \(x^2 = 64\)?

Question 4 of 5

Which of the following is an irrational number?

Question 5 of 5

What is \(\sqrt{\dfrac{36}{81}}\)?