IB AA ยท 12A Laws of Exponents
A negative exponent turns into 1 over the same base with a positive exponent. Flip first, then take the root.
Jump to section:
Negative means reciprocal
The formula is
\(a^{-n}=\dfrac{1}{a^n}\)
A negative exponent turns into 1 divided by the same expression with a positive exponent.
\(36^{-\frac{1}{2}}=\dfrac{1}{36^{\frac{1}{2}}}\)
\(\dfrac{1}{6}\)
Split, then flip
The formula is
\(a^{-\frac{m}{n}}=\dfrac{1}{\left(a^{\frac{1}{n}}\right)^m}\)
Take the root first, raise to the remaining power, then write 1 over that value.
A fraction to a negative power
A negative exponent flips the fraction. Then apply the positive power.
\(\left(\dfrac{a}{b}\right)^{-n}=\left(\dfrac{b}{a}\right)^{n}\)
\(\left(\dfrac{9}{4}\right)^{-\frac{1}{2}}=\left(\dfrac{4}{9}\right)^{\frac{1}{2}}\)
\(\dfrac{2}{3}\)
Small perfect powers
Take the root first so the numbers stay small. Know these by heart.
| Power | Value | Power | Value |
|---|---|---|---|
| \(2^2\) | 4 | \(3^2\) | 9 |
| \(2^3\) | 8 | \(3^3\) | 27 |
| \(2^4\) | 16 | \(3^4\) | 81 |
| \(2^5\) | 32 | \(5^2\) | 25 |
| \(4^3\) | 64 | \(5^3\) | 125 |
A unit fraction with a negative sign. Flip, then take the root.
The formula is.\(a^{-n}=\dfrac{1}{a^n}\)
\(36^{-\frac{1}{2}}=\dfrac{1}{36^{\frac{1}{2}}}\)
The formula is.\(a^{\frac{1}{n}}=\sqrt[n]{a}\)
\(36^{\frac{1}{2}}=\sqrt{36}\)
\(\sqrt{36}=6\)
\(\dfrac{1}{6}\)
Answer: 16
The formula is.\(a^{-n}=\dfrac{1}{a^n}\)
\(81^{-\frac{1}{2}}=\dfrac{1}{81^{\frac{1}{2}}}\)
The formula is.\(a^{\frac{1}{n}}=\sqrt[n]{a}\)
\(81^{\frac{1}{2}}=\sqrt{81}\)
\(\sqrt{81}=9\)
\(\dfrac{1}{9}\)
Answer: 19
The formula is.\(a^{-n}=\dfrac{1}{a^n}\)
\(8^{-\frac{1}{3}}=\dfrac{1}{8^{\frac{1}{3}}}\)
The formula is.\(a^{\frac{1}{n}}=\sqrt[n]{a}\)
\(8^{\frac{1}{3}}=\sqrt[3]{8}\)
\(\sqrt[3]{8}=2\)
\(\dfrac{1}{2}\)
Answer: 12
The formula is.\(a^{-n}=\dfrac{1}{a^n}\)
\(16^{-\frac{1}{4}}=\dfrac{1}{16^{\frac{1}{4}}}\)
The formula is.\(a^{\frac{1}{n}}=\sqrt[n]{a}\)
\(16^{\frac{1}{4}}=\sqrt[4]{16}\)
\(\sqrt[4]{16}=2\)
\(\dfrac{1}{2}\)
Answer: 12
The formula is.\(a^{-n}=\dfrac{1}{a^n}\)
\(64^{-\frac{1}{2}}=\dfrac{1}{64^{\frac{1}{2}}}\)
The formula is.\(a^{\frac{1}{n}}=\sqrt[n]{a}\)
\(64^{\frac{1}{2}}=\sqrt{64}\)
\(\sqrt{64}=8\)
\(\dfrac{1}{8}\)
Answer: 18
The formula is.\(a^{-n}=\dfrac{1}{a^n}\)
\(125^{-\frac{1}{3}}=\dfrac{1}{125^{\frac{1}{3}}}\)
The formula is.\(a^{\frac{1}{n}}=\sqrt[n]{a}\)
\(125^{\frac{1}{3}}=\sqrt[3]{125}\)
\(\sqrt[3]{125}=5\)
\(\dfrac{1}{5}\)
Answer: 15
Flip first. Take the root, then raise to the remaining power.
The formula is.\(a^{-n}=\dfrac{1}{a^n}\)
\(32^{-\frac{3}{5}}=\dfrac{1}{32^{\frac{3}{5}}}\)
The formula is.\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^m\)
\(32^{\frac{3}{5}}=\left(32^{\frac{1}{5}}\right)^{3}\)
\(32^{\frac{1}{5}}=2\)
\(2^{3}=8\)
\(\dfrac{1}{8}\)
Answer: 18
The formula is.\(a^{-n}=\dfrac{1}{a^n}\)
\(64^{-\frac{2}{3}}=\dfrac{1}{64^{\frac{2}{3}}}\)
The formula is.\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^m\)
\(64^{\frac{2}{3}}=\left(64^{\frac{1}{3}}\right)^{2}\)
\(64^{\frac{1}{3}}=4\)
\(4^{2}=16\)
\(\dfrac{1}{16}\)
Answer: 116
The formula is.\(a^{-n}=\dfrac{1}{a^n}\)
\(27^{-\frac{4}{3}}=\dfrac{1}{27^{\frac{4}{3}}}\)
The formula is.\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^m\)
\(27^{\frac{4}{3}}=\left(27^{\frac{1}{3}}\right)^{4}\)
\(27^{\frac{1}{3}}=3\)
\(3^{4}=81\)
\(\dfrac{1}{81}\)
Answer: 181
The formula is.\(a^{-n}=\dfrac{1}{a^n}\)
\(81^{-\frac{3}{4}}=\dfrac{1}{81^{\frac{3}{4}}}\)
The formula is.\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^m\)
\(81^{\frac{3}{4}}=\left(81^{\frac{1}{4}}\right)^{3}\)
\(81^{\frac{1}{4}}=3\)
\(3^{3}=27\)
\(\dfrac{1}{27}\)
Answer: 127
The formula is.\(a^{-n}=\dfrac{1}{a^n}\)
\(125^{-\frac{2}{3}}=\dfrac{1}{125^{\frac{2}{3}}}\)
The formula is.\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^m\)
\(125^{\frac{2}{3}}=\left(125^{\frac{1}{3}}\right)^{2}\)
\(125^{\frac{1}{3}}=5\)
\(5^{2}=25\)
\(\dfrac{1}{25}\)
Answer: 125
The formula is.\(a^{-n}=\dfrac{1}{a^n}\)
\(16^{-\frac{3}{4}}=\dfrac{1}{16^{\frac{3}{4}}}\)
The formula is.\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^m\)
\(16^{\frac{3}{4}}=\left(16^{\frac{1}{4}}\right)^{3}\)
\(16^{\frac{1}{4}}=2\)
\(2^{3}=8\)
\(\dfrac{1}{8}\)
Answer: 18
A fraction to a negative rational power. Flip the fraction, then take roots.
The formula is.\(\left(\dfrac{a}{b}\right)^{-n}=\left(\dfrac{b}{a}\right)^{n}\)
\(\left(\dfrac{9}{4}\right)^{-\frac{1}{2}}=\left(\dfrac{4}{9}\right)^{\frac{1}{2}}\)
\(\dfrac{4^{\frac{1}{2}}}{9^{\frac{1}{2}}}=\dfrac{2}{3}\)
Answer: 23
\(\left(\dfrac{1}{9}\right)^{-\frac{1}{2}}=\left(\dfrac{9}{1}\right)^{\frac{1}{2}}\)
\(9^{\frac{1}{2}}=3\)
Answer: 3
The formula is.\(\left(\dfrac{a}{b}\right)^{-n}=\left(\dfrac{b}{a}\right)^{n}\)
\(\left(\dfrac{8}{27}\right)^{-\frac{2}{3}}=\left(\dfrac{27}{8}\right)^{\frac{2}{3}}\)
The formula is.\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^m\)
\(\left(\dfrac{27}{8}\right)^{\frac{2}{3}}=\left(\dfrac{27^{\frac{1}{3}}}{8^{\frac{1}{3}}}\right)^{2}\)
\(27^{\frac{1}{3}}=3\)
\(8^{\frac{1}{3}}=2\)
\(\left(\dfrac{3}{2}\right)^{2}=\dfrac{9}{4}\)
Answer: 94
The formula is.\(\left(\dfrac{a}{b}\right)^{-n}=\left(\dfrac{b}{a}\right)^{n}\)
\(\left(\dfrac{4}{9}\right)^{-\frac{3}{2}}=\left(\dfrac{9}{4}\right)^{\frac{3}{2}}\)
The formula is.\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^m\)
\(\left(\dfrac{9}{4}\right)^{\frac{3}{2}}=\left(\dfrac{9^{\frac{1}{2}}}{4^{\frac{1}{2}}}\right)^{3}\)
\(9^{\frac{1}{2}}=3\)
\(4^{\frac{1}{2}}=2\)
\(\left(\dfrac{3}{2}\right)^{3}=\dfrac{27}{8}\)
Answer: 278
The formula is.\(\left(\dfrac{a}{b}\right)^{-n}=\left(\dfrac{b}{a}\right)^{n}\)
\(\left(\dfrac{27}{8}\right)^{-\frac{2}{3}}=\left(\dfrac{8}{27}\right)^{\frac{2}{3}}\)
The formula is.\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^m\)
\(\left(\dfrac{8}{27}\right)^{\frac{2}{3}}=\left(\dfrac{8^{\frac{1}{3}}}{27^{\frac{1}{3}}}\right)^{2}\)
\(8^{\frac{1}{3}}=2\)
\(27^{\frac{1}{3}}=3\)
\(\left(\dfrac{2}{3}\right)^{2}=\dfrac{4}{9}\)
Answer: 49
The formula is.\(\left(\dfrac{a}{b}\right)^{-n}=\left(\dfrac{b}{a}\right)^{n}\)
\(\left(\dfrac{16}{81}\right)^{-\frac{3}{4}}=\left(\dfrac{81}{16}\right)^{\frac{3}{4}}\)
The formula is.\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^m\)
\(\left(\dfrac{81}{16}\right)^{\frac{3}{4}}=\left(\dfrac{81^{\frac{1}{4}}}{16^{\frac{1}{4}}}\right)^{3}\)
\(81^{\frac{1}{4}}=3\)
\(16^{\frac{1}{4}}=2\)
\(\left(\dfrac{3}{2}\right)^{3}=\dfrac{27}{8}\)
Answer: 278
Need help with negative rational exponents?
Book a Session with Amy