Simplifying Radicals

A radical is the mathematical symbol for a root — most commonly a square root ($\sqrt{\phantom{x}}$). The expression $\sqrt{49}$ asks “what number multiplied by itself gives 49?” and the answer is 7. But what about $\sqrt{50}$? There is no whole number whose square is 50, so you cannot evaluate it to a neat integer. Instead, you simplify it: $\sqrt{50} = 5\sqrt{2}$. Learning to simplify radicals is essential for solving equations, working with the Pythagorean theorem, and handling formulas throughout algebra and the trades.

Perfect Squares Review

Before simplifying, you need to recognize perfect squares — numbers that are the square of a whole number:

$1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, \ldots$

Their square roots are clean whole numbers:

$\sqrt{1} = 1, \quad \sqrt{4} = 2, \quad \sqrt{9} = 3, \quad \sqrt{16} = 4, \quad \sqrt{25} = 5, \ldots$

If the number under the radical is a perfect square, you are done. If not, you simplify by finding the largest perfect square factor.

The Product Property of Radicals

The key property used for simplifying is:

$\sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \quad \text{(where } a \geq 0 \text{ and } b \geq 0\text{)}$

This works in reverse too: $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$. The strategy for simplifying is to factor the radicand (the number under the radical) so that one factor is the largest possible perfect square.

Simplifying Numerical Radicals

Example 1: Simplify $\sqrt{72}$

Step 1 — Find the largest perfect square factor of 72.

Factor 72: $72 = 36 \times 2$. Since $36 = 6^2$, it is a perfect square.

Step 2 — Apply the product property:

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

Answer: $6\sqrt{2}$

Example 2: Simplify $\sqrt{200}$

Find the largest perfect square factor: $200 = 100 \times 2$, and $100 = 10^2$.

$\sqrt{200} = \sqrt{100 \times 2} = \sqrt{100} \cdot \sqrt{2} = 10\sqrt{2}$

Answer: $10\sqrt{2}$

Example 3: Simplify $\sqrt{48}$

$48 = 16 \times 3$, and $16 = 4^2$.

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

Answer: $4\sqrt{3}$

Example 4: Simplify $\sqrt{180}$

$180 = 36 \times 5$.

$\sqrt{180} = \sqrt{36 \times 5} = 6\sqrt{5}$

Answer: $6\sqrt{5}$

Systematic Method: Prime Factorization

When the largest perfect square factor is not obvious, use prime factorization.

Example 5: Simplify $\sqrt{252}$

Step 1 — Prime factorize:

$252 = 2^2 \times 3^2 \times 7$

Step 2 — For each pair of identical factors, bring one copy outside the radical:

$\sqrt{2^2 \times 3^2 \times 7} = 2 \cdot 3 \cdot \sqrt{7} = 6\sqrt{7}$

Answer: $6\sqrt{7}$

The rule: for every pair of identical prime factors under the radical, one copy comes out; unpaired factors stay inside.

Simplifying Radicals with Variables

The same product property applies to variables. The key idea is that $\sqrt{x^2} = |x|$, but in algebra courses where variables represent positive quantities (or we are dealing with even powers), we typically write $\sqrt{x^2} = x$.

Rule for variable exponents under a square root:

• Divide the exponent by 2.
• The quotient comes outside the radical; the remainder stays inside.

Example 6: Simplify $\sqrt{x^6}$

$6 \div 2 = 3$ with remainder 0. Everything comes out:

$\sqrt{x^6} = x^3$

Answer: $x^3$

Example 7: Simplify $\sqrt{x^7}$

$7 \div 2 = 3$ with remainder 1. Three copies come out; one stays inside:

$\sqrt{x^7} = x^3\sqrt{x}$

Answer: $x^3\sqrt{x}$

Example 8: Simplify $\sqrt{50x^4y^3}$

Separate into numbers and variables:

$\sqrt{50x^4y^3} = \sqrt{50} \cdot \sqrt{x^4} \cdot \sqrt{y^3}$

Simplify each piece:

• $\sqrt{50} = 5\sqrt{2}$
• $\sqrt{x^4} = x^2$ (exponent $4 \div 2 = 2$)
• $\sqrt{y^3} = y\sqrt{y}$ (exponent $3 \div 2 = 1$ remainder $1$)

Combine:

$\sqrt{50x^4y^3} = 5x^2 y\sqrt{2y}$

Answer: $5x^2 y\sqrt{2y}$

Example 9: Simplify $\sqrt{72a^5b^2}$

• $\sqrt{72} = 6\sqrt{2}$
• $\sqrt{a^5} = a^2\sqrt{a}$
• $\sqrt{b^2} = b$

$\sqrt{72a^5b^2} = 6a^2 b\sqrt{2a}$

Answer: $6a^2 b\sqrt{2a}$

Simplifying with Coefficients Already Outside

Sometimes the expression already has a coefficient in front of the radical.

Example 10: Simplify $3\sqrt{75}$

First simplify $\sqrt{75}$: $75 = 25 \times 3$, so $\sqrt{75} = 5\sqrt{3}$.

Then multiply:

$3\sqrt{75} = 3 \cdot 5\sqrt{3} = 15\sqrt{3}$

Answer: $15\sqrt{3}$

Cube Roots — A Brief Introduction

A cube root asks “what number cubed gives this value?” The notation is $\sqrt[3]{a}$.

$\sqrt[3]{8} = 2 \quad \text{because } 2^3 = 8$

$\sqrt[3]{27} = 3 \quad \text{because } 3^3 = 27$

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

Unlike square roots, cube roots can have negative radicands because a negative number cubed is negative.

The product property still works: $\sqrt[3]{ab} = \sqrt[3]{a} \cdot \sqrt[3]{b}$.

Example 11: Simplify $\sqrt[3]{54}$

$54 = 27 \times 2$, and $27 = 3^3$.

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

Answer: $3\sqrt[3]{2}$

Example 12: Simplify $\sqrt[3]{-128}$

$-128 = -1 \times 128 = -1 \times 64 \times 2$, and $64 = 4^3$.

$\sqrt[3]{-128} = \sqrt[3]{-64 \times 2} = \sqrt[3]{-64} \cdot \sqrt[3]{2} = -4\sqrt[3]{2}$

Answer: $-4\sqrt[3]{2}$

Real-World Application: Electrician — Wire Gauge and Cross-Sectional Area

When electricians calculate the cross-sectional area of a wire from its diameter, they use:

$A = \frac{\pi d^2}{4}$

Rearranging to find the diameter from the area gives:

$d = \sqrt{\frac{4A}{\pi}} = \frac{2\sqrt{A}}{\sqrt{\pi}}$

If the required area is 50 square millimeters:

$d = \frac{2\sqrt{50}}{\sqrt{\pi}} = \frac{2 \cdot 5\sqrt{2}}{\sqrt{\pi}} = \frac{10\sqrt{2}}{\sqrt{\pi}} \approx \frac{10 \times 1.414}{1.772} \approx 7.98 \text{ mm}$

Being able to simplify $\sqrt{50}$ to $5\sqrt{2}$ makes the intermediate algebra cleaner and reduces the risk of calculator error.

Real-World Application: Carpentry — Diagonal of a Rectangle

A carpenter needs to find the diagonal of a rectangular deck that is 6 feet by 10 feet. Using the Pythagorean theorem:

$d = \sqrt{6^2 + 10^2} = \sqrt{36 + 100} = \sqrt{136}$

Simplify: $136 = 4 \times 34$.

$d = \sqrt{4 \times 34} = 2\sqrt{34} \approx 2(5.831) \approx 11.66 \text{ feet}$

Leaving the answer as $2\sqrt{34}$ is the exact form. The decimal $\approx 11.66$ feet is the approximate form used for cutting.

Common Mistakes to Avoid

1. Not using the largest perfect square factor. $\sqrt{72} = \sqrt{4 \times 18} = 2\sqrt{18}$ is technically correct but not fully simplified. You should continue: $\sqrt{18} = 3\sqrt{2}$, giving $6\sqrt{2}$. Better to find the largest perfect square factor (36) from the start.
2. Forgetting that $\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$. The product property works for multiplication: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$. There is no corresponding rule for addition. For example, $\sqrt{9 + 16} = \sqrt{25} = 5$, but $\sqrt{9} + \sqrt{16} = 3 + 4 = 7$.
3. Pulling out the wrong number of variable copies. For $\sqrt{x^5}$, divide 5 by 2 to get 2 remainder 1: $x^2\sqrt{x}$, not $x^5$ or $x^{2.5}$.
4. Forgetting to simplify the coefficient outside. After simplifying the radical, always multiply any existing coefficient by the number that came out of the radical.
5. Treating cube roots like square roots. For cube roots, you need groups of three identical factors, not two.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Simplifying a radical means rewriting it so the number under the radical has no perfect square (or perfect cube) factors
• The product property $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ is the core tool for simplifying
• For numerical radicals, find the largest perfect square factor or use prime factorization
• For variable radicals, divide the exponent by 2 — the quotient comes out, the remainder stays in
• There is no sum property — $\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$
• Cube roots use groups of three and can handle negative radicands

Return to Algebra for more topics in this section.