Before you read

Answer first — see what you already know.

A surveyor measures a diagonal distance of $75$ meters. What is this in simplified radical form?

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Answer:

Before you read

Answer first — see what you already know.

In a scientific formula, $x^{\frac{2}{3}}$ appears. Which radical expression is equivalent?

Check your answer

Answer:

Theory

Radicals and Exponents: The Connection

Radicals and rational exponents are two notations for the same thing:

$\sqrt{x} = x^{\frac{1}{2}} \qquad \sqrt[3]{x} = x^{\frac{1}{3}} \qquad \sqrt[n]{x} = x^{\frac{1}{n}}$

$\sqrt[n]{x^m} = x^{\frac{m}{n}} \qquad (\sqrt[n]{x})^m = x^{\frac{m}{n}}$

Converting between radicals and rational exponents

The denominator of the exponent is the root, and the numerator is the power.

All exponent rules from B1.1 apply to rational exponents too!

Theory

Simplifying Radicals

To simplify $\sqrt{n}$ : find the largest perfect square that divides $n$ .

$\sqrt{n} = \sqrt{a^2 \cdot b} = a\sqrt{b}$

Perfect squares to know: $4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144$ .

Example 1

Simplify $\sqrt{50}$

$50 = 25 \times 2$

$\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$

5\sqrt{2}

Example 2

Simplify $\sqrt{72}$

$72 = 36 \times 2$

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

6\sqrt{2}

Example 3

Simplify $\sqrt{x^6}$

$\sqrt{x^6} = (x^6)^{\frac{1}{2}} = x^3$

(assuming $x \geq 0$ )

x^3

Example 4

Simplify $\sqrt{18x^4}$

$\sqrt{18x^4} = \sqrt{9 \cdot 2 \cdot x^4}$

$= 3x^2\sqrt{2}$

3x^2\sqrt{2}

3 more sections including examples, practice problems, and step-by-step solutions.

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