Evaluate $27^{\frac{2}{3}}$, $16^{-\frac{3}{4}}$ and $\left(\frac{4}{9}\right)^{-\frac{1}{2}}$.

$9$, $\frac{1}{8}$ and $\frac{3}{2}$.

Grade 11 · MCR3U · Functions

MCR3U: Exponent Laws and Rational Exponents

Every exponent law comes from what a power means: $a^n$ is $a$ multiplied by itself $n$ times. Count the factors and the laws follow, and the same counting decides what zero, negative and fractional exponents have to mean.

How It Works

Products and quotients. $a^m \times a^n = a^{m + n}$ , because you are counting all the factors together. $a^m \div a^n = a^{m - n}$ , because the shared factors cancel. Both laws need the same base.

Power of a power. $(a^m)^n = a^{mn}$ . A power of a product or a quotient applies to every factor inside: $(2x^3)^4 = 16x^{12}$ , not $2x^{12}$ .

Zero exponent. $a^0 = 1$ for $a \ne 0$ , since $\frac{a^n}{a^n} = a^{n - n}$ has to equal $1$ .

Negative exponent. $a^{-n} = \frac{1}{a^n}$ . A negative exponent means reciprocal, not a negative number: $2^{-3} = \frac{1}{8}$ . A factor can move across the fraction bar if its exponent changes sign, which is how $\frac{4^{-3}}{7^{-2}}$ becomes $\frac{7^2}{4^3}$ .

Rational exponents. $a^{\frac{1}{n}} = \sqrt[n]{a}$ , and $a^{\frac{m}{n}} = (\sqrt[n]{a})^m$ . The denominator is the root and the numerator is the power. Take the root first to keep the numbers small: $32^{\frac{3}{5}} = 2^3 = 8$ .

A routine for long expressions. Turn every root into a rational exponent. Simplify inside the brackets, apply the outside powers, then combine like bases. Finish with positive exponents only, unless the question says otherwise.

Common slips. $-2^4 = -16$ but $(-2)^4 = 16$ , because the exponent binds before the minus sign. And there is no law for sums: $(a + b)^2$ is not $a^2 + b^2$ .

5 Worked Examples

Easy first, test level last. Try each one on paper, then open the solution.

Easy
Simplify $\frac{x^5 \times x^{-2}}{x^7}$ , using positive exponents.
Show solution
1. Top: $x^{5 + (-2)} = x^3$ .
2. Divide: $x^{3 - 7} = x^{-4}$ , then take the reciprocal.
Answer: $\frac{1}{x^4}$
Easy
Evaluate $27^{\frac{2}{3}}$ , $16^{-\frac{3}{4}}$ and $\left(\frac{4}{9}\right)^{-\frac{1}{2}}$ .
Show solution
1. $27^{\frac{2}{3}} = (\sqrt[3]{27})^2 = 3^2 = 9$ .
2. $16^{-\frac{3}{4}} = \frac{1}{(\sqrt[4]{16})^3} = \frac{1}{2^3} = \frac{1}{8}$ .
3. $\left(\frac{4}{9}\right)^{-\frac{1}{2}} = \left(\frac{9}{4}\right)^{\frac{1}{2}} = \frac{3}{2}$ .
Answer: $9$ , $\frac{1}{8}$ and $\frac{3}{2}$ .
Medium
Simplify $(3a^2b^{-1})^3(a^{-1}b)^2$ , using positive exponents.
Show solution
1. Apply the outside powers: $27a^6b^{-3} \times a^{-2}b^2$ .
2. Combine like bases: $27a^{6 - 2}b^{-3 + 2} = 27a^4b^{-1}$ .
Answer: $\frac{27a^4}{b}$
Medium
Write $\frac{\sqrt[3]{x^7}}{\sqrt{x}}$ as a single power of $x$ .
Show solution
1. Turn the roots into exponents: $\frac{x^{\frac{7}{3}}}{x^{\frac{1}{2}}}$ .
2. Subtract: $\frac{7}{3} - \frac{1}{2} = \frac{14}{6} - \frac{3}{6} = \frac{11}{6}$ .
Answer: $x^{\frac{11}{6}}$
Test level
Simplify $\frac{(8x^6y^{-3})^{\frac{2}{3}}}{(16x^{-2}y^4)^{\frac{1}{2}}}$ , using positive exponents.
Show solution
1. Top: $8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 4$ , so the top is $4x^4y^{-2}$ .
2. Bottom: $16^{\frac{1}{2}} = 4$ , so the bottom is $4x^{-1}y^2$ .
3. Divide: $\frac{4}{4}x^{4 - (-1)}y^{-2 - 2} = x^5y^{-4}$ .
Answer: $\frac{x^5}{y^4}$

Learn It on Martian Lab

Martian Lab teaches this in the Exponent Laws lesson of its MCR3U Exponential Functions unit. Watch it, then take the Exponential Functions practice tests to see if it stuck.

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Free Exponential Functions practice test, with solutions

More Exponential Functions topics: Graphing Exponential Functions · Exponential Growth and Decay

More MCR3U topics: Inverse Functions · Transformations of Functions · Sine Law and the Ambiguous Case · Domain, Range and Function Notation · Simplifying Rational Expressions · Operations with Rational Expressions · Max/Min and Vertex Form · Linear-Quadratic Systems · Quadratic Word Problems · Simplifying Radicals · Special Angles and the CAST Rule · Cosine Law and 3D Problems · Transformations of Sine and Cosine · Sinusoidal Word Problems · Arithmetic Sequences and Series · Geometric Sequences and Series · Pascal's Triangle and the Binomial Theorem · Simple and Compound Interest · Annuities: Future and Present Value

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