Law of Rational Exponents

Law of Rational Exponents

When a base is raised to a rational/fraction exponent, the numerator is the exponent and the denominator is the index of a radical/root.

x^{m/n}

\sqrt[n]{x^m}

\Big(\sqrt[n]{x}\Big)^m

Exponent Form

Root Form

Root Form

If the index isn’t shown, it is implied to be a 2. \sqrt{x}=x^{1/2}

When evaluating, it is generally better to put the expression into root form. 32^{2/5}=(\sqrt[5]{x})^2=(2)^2=4

In case you don’t have a calculator, it’s usually easier to evaluate the root before the exponent.

When simplifying, it is generally better to put the expression into exponent form. This will make it easier to use exponent laws.\frac{\sqrt[3]{x^4}}{\sqrt{x^3}}=\frac{x^{4/3}}{x^{3/2}}=x^{4/3-3/2}=x^{-1/6}=\frac{1}{x^{1/6}}=\frac{1}{\sqrt[6]{x}}

In general, return any answer to the original form of the question.

Practice

EXAMPLE 1

Simplify then evaluate: 2^{11} \div 2^8

SOLUTION

Using the Law of Division: 2^{11} \div 2^8=2^{11-8}=2^3

Evaluating: 2^3=8

EXAMPLE 2

Simplify: (m^0)^{10}

SOLUTION

Using the Law of Powers: (m^0)^{10}=m^0

We should more properly use the Law of Zero Exponents to write: m^0=1

EXAMPLE 3

Find the value of x that makes the equation true: 4^x \times 4^4=4^7

SOLUTION

Since we are multiplying the two powers, we use the Law of Multiplication: x+4=7

Therefore: x=3

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