Square Roots and Cube Roots — How to Find and Simplify

Grade: 7-8 | Topic: Arithmetic

What You Will Learn

After reading this page you will understand what square roots and cube roots are, know all the perfect squares and cubes worth memorizing, and be able to simplify radical expressions like $\sqrt{72}$ or $\sqrt[3]{54}$ with confidence. Roots are the inverse of exponents, so mastering them rounds out your understanding of powers.

Theory

What is a square root?

The square root of a number $n$ is the value that, when multiplied by itself, equals $n$:

$\sqrt{n} = x \quad \text{means} \quad x^{2} = n$

For example, $\sqrt{49} = 7$ because $7 \times 7 = 49$.

Every positive number has two square roots — one positive and one negative. The radical symbol $\sqrt{\phantom{x}}$ always refers to the principal (positive) root. If you need both roots, write $\pm\sqrt{n}$.

Perfect squares you should memorize

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

Knowing these instantly makes simplifying radicals much faster.

How to simplify a square root

If the number under the radical is not a perfect square, simplify it by factoring out the largest perfect square:

$\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$

Strategy: Find the largest perfect square factor of the number, take its root, and leave the rest under the radical.

Example: Simplify $\sqrt{72}$.

$72 = 36 \times 2 \implies \sqrt{72} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}$

What is a cube root?

The cube root of a number $n$ is the value that, when multiplied by itself three times, equals $n$:

$\sqrt[3]{n} = x \quad \text{means} \quad x^{3} = n$

For example, $\sqrt[3]{27} = 3$ because $3 \times 3 \times 3 = 27$.

Unlike square roots, cube roots can be negative: $\sqrt[3]{-8} = -2$ because $(-2)^{3} = -8$.

Perfect cubes you should memorize

$n$	1	8	27	64	125	216	343	512	729	1000
$\sqrt[3]{n}$	1	2	3	4	5	6	7	8	9	10

How to simplify a cube root

The same factoring strategy works — find the largest perfect cube factor:

$\sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b}$

Example: Simplify $\sqrt[3]{54}$.

$54 = 27 \times 2 \implies \sqrt[3]{54} = \sqrt[3]{27} \cdot \sqrt[3]{2} = 3\sqrt[3]{2}$

Connection to exponents

Roots are fractional exponents:

$\sqrt{a} = a^{1/2} \qquad \sqrt[3]{a} = a^{1/3} \qquad \sqrt[n]{a} = a^{1/n}$

This means all exponent rules apply to roots as well, which is why this topic fits under the exponents umbrella.

Worked Examples

Example 1: Evaluating a perfect square root

Problem: Find $\sqrt{144}$.

Step 1: Ask: what number times itself equals 144? $12 \times 12 = 144$

Answer: $\sqrt{144} = \mathbf{12}$

Example 2: Simplifying a non-perfect square root

Problem: Simplify $\sqrt{200}$.

Step 1: Find the largest perfect square factor of 200. $200 = 100 \times 2$

Step 2: Split and simplify. $\sqrt{200} = \sqrt{100} \cdot \sqrt{2} = 10\sqrt{2}$

Answer: $\sqrt{200} = \mathbf{10\sqrt{2}}$

Example 3: Simplifying a cube root

Problem: Simplify $\sqrt[3]{128}$.

Step 1: Find the largest perfect cube factor of 128. $128 = 64 \times 2$

Step 2: Split and simplify. $\sqrt[3]{128} = \sqrt[3]{64} \cdot \sqrt[3]{2} = 4\sqrt[3]{2}$

Answer: $\sqrt[3]{128} = \mathbf{4\sqrt[3]{2}}$

Example 4: Cube root of a negative number

Problem: Evaluate $\sqrt[3]{-216}$.

Step 1: Determine what number cubed gives $-216$. $(-6)^{3} = (-6)(-6)(-6) = 36 \times (-6) = -216$

Answer: $\sqrt[3]{-216} = \mathbf{-6}$

Example 5: Simplifying a square root with variables

Problem: Simplify $\sqrt{50x^{4}}$.

Step 1: Factor the number and variable parts. $50 = 25 \times 2, \qquad x^{4} = (x^{2})^{2}$

Step 2: Take the square root of each perfect square factor. $\sqrt{50x^{4}} = \sqrt{25} \cdot \sqrt{2} \cdot \sqrt{(x^{2})^{2}} = 5 \cdot \sqrt{2} \cdot x^{2}$

Answer: $\sqrt{50x^{4}} = \mathbf{5x^{2}\sqrt{2}}$

Common Mistakes

Mistake 1: Thinking the square root of a sum equals the sum of the square roots

❌ $\sqrt{9 + 16} = \sqrt{9} + \sqrt{16} = 3 + 4 = 7$

✅ $\sqrt{9 + 16} = \sqrt{25} = 5$

Why this matters: The product property $\sqrt{ab} = \sqrt{a}\sqrt{b}$ works for multiplication, but there is no equivalent rule for addition. This is one of the most common radical errors.

Mistake 2: Not fully simplifying the radical

❌ $\sqrt{48} = 2\sqrt{12}$ (stopped too early)

✅ $\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$

Why this matters: If you use a smaller perfect square factor (like 4 instead of 16), you will need to simplify again. Always look for the largest perfect square factor to save time.

Mistake 3: Forgetting that cube roots can be negative

❌ $\sqrt[3]{-64}$ = "undefined" or "error"

✅ $\sqrt[3]{-64} = -4$ because $(-4)^{3} = -64$

Why this matters: Unlike even roots, odd roots (cube roots, fifth roots, etc.) accept negative inputs and produce negative outputs. Cube roots of negative numbers are perfectly real.

Practice Problems

Try these on your own before checking the answers:

1. Evaluate $\sqrt{196}$.
2. Simplify $\sqrt{180}$.
3. Evaluate $\sqrt[3]{343}$.
4. Simplify $\sqrt[3]{250}$.
5. Simplify $\sqrt{75x^{6}}$.

Click to see answers

1. $\sqrt{196} = 14$ (because $14 \times 14 = 196$)
2. $\sqrt{180} = \sqrt{36 \times 5} = 6\sqrt{5}$
3. $\sqrt[3]{343} = 7$ (because $7^{3} = 343$)
4. $\sqrt[3]{250} = \sqrt[3]{125 \times 2} = 5\sqrt[3]{2}$
5. $\sqrt{75x^{6}} = \sqrt{25 \times 3} \cdot \sqrt{(x^{3})^{2}} = 5x^{3}\sqrt{3}$

Summary

• A square root asks "what squared gives this number?" and a cube root asks "what cubed gives this number?"
• Memorize perfect squares (1 through 144) and perfect cubes (1 through 1000) to work faster.
• To simplify a radical, factor out the largest perfect square (or cube) and take its root.
• $\sqrt{a \cdot b} = \sqrt{a}\sqrt{b}$ works for products, but not for sums.
• Cube roots can handle negative numbers; square roots (in real numbers) cannot.
• Roots are fractional exponents: $\sqrt{a} = a^{1/2}$, $\sqrt[3]{a} = a^{1/3}$.

Related Topics

• Exponents and Powers — Rules, Examples, and Practice
• Exponent Rules — All Laws of Exponents Explained
• Negative Exponents — How to Simplify and Solve

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