IB AA ยท 12A Laws of Exponents
A unit fraction is a root. For a power m over n, take the nth root first, then raise to the m.
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A root is a fraction
The formula is
\(a^{\frac{1}{n}}=\sqrt[n]{a}\)
\(36^{\frac{1}{2}}=\sqrt{36}\)
6
Split the exponent
The formula is
\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^m\)
Take the root first so the numbers stay small.
\(27^{\frac{2}{3}}=\left(27^{\frac{1}{3}}\right)^2\)
\(3^2=9\)
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 fraction raised to a power
The outer exponent hits the numerator and the denominator.
\(\left(\dfrac{a}{b}\right)^{\frac{m}{n}}=\dfrac{a^{\frac{m}{n}}}{b^{\frac{m}{n}}}\)
The exponent is a unit fraction. Write it as a root, then evaluate.
The formula is.\(a^{\frac{1}{n}}=\sqrt[n]{a}\)
\(36^{\frac{1}{2}}=\sqrt{36}\)
\(\sqrt{36}=6\)
Answer: 6
The formula is.\(a^{\frac{1}{n}}=\sqrt[n]{a}\)
\(81^{\frac{1}{2}}=\sqrt{81}\)
\(\sqrt{81}=9\)
Answer: 9
The formula is.\(a^{\frac{1}{n}}=\sqrt[n]{a}\)
\(64^{\frac{1}{2}}=\sqrt{64}\)
\(\sqrt{64}=8\)
Answer: 8
The formula is.\(a^{\frac{1}{n}}=\sqrt[n]{a}\)
\(125^{\frac{1}{3}}=\sqrt[3]{125}\)
\(\sqrt[3]{125}=5\)
Answer: 5
The formula is.\(a^{\frac{1}{n}}=\sqrt[n]{a}\)
\(64^{\frac{1}{3}}=\sqrt[3]{64}\)
\(\sqrt[3]{64}=4\)
Answer: 4
The formula is.\(a^{\frac{1}{n}}=\sqrt[n]{a}\)
\(81^{\frac{1}{4}}=\sqrt[4]{81}\)
\(\sqrt[4]{81}=3\)
Answer: 3
Take the root first, then raise to the remaining power.
The formula is.\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^m\)
\(27^{\frac{2}{3}}=\left(27^{\frac{1}{3}}\right)^{2}\)
\(27^{\frac{1}{3}}=3\)
\(3^2=9\)
Answer: 9
The formula is.\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^m\)
\(32^{\frac{3}{5}}=\left(32^{\frac{1}{5}}\right)^{3}\)
\(32^{\frac{1}{5}}=2\)
\(2^3=8\)
Answer: 8
The formula is.\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^m\)
\(81^{\frac{3}{4}}=\left(81^{\frac{1}{4}}\right)^{3}\)
\(81^{\frac{1}{4}}=3\)
\(3^3=27\)
Answer: 27
The formula is.\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^m\)
\(64^{\frac{2}{3}}=\left(64^{\frac{1}{3}}\right)^{2}\)
\(64^{\frac{1}{3}}=4\)
\(4^2=16\)
Answer: 16
The formula is.\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^m\)
\(125^{\frac{4}{3}}=\left(125^{\frac{1}{3}}\right)^{4}\)
\(125^{\frac{1}{3}}=5\)
\(5^4=625\)
Answer: 625
The formula is.\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^m\)
\(16^{\frac{5}{4}}=\left(16^{\frac{1}{4}}\right)^{5}\)
\(16^{\frac{1}{4}}=2\)
\(2^5=32\)
Answer: 32
A fraction base, or a larger leftover power. Take each root first.
The formula is.\(\left(\dfrac{a}{b}\right)^{\frac{m}{n}}=\dfrac{a^{\frac{m}{n}}}{b^{\frac{m}{n}}}\)
\(\left(\dfrac{25}{9}\right)^{\frac{3}{2}}=\dfrac{25^{\frac{3}{2}}}{9^{\frac{3}{2}}}\)
The formula is.\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^m\)
\(25^{\frac{3}{2}}=\left(25^{\frac{1}{2}}\right)^3\)
\(25^{\frac{1}{2}}=5\)
\(5^3=125\)
\(9^{\frac{3}{2}}=\left(9^{\frac{1}{2}}\right)^3\)
\(9^{\frac{1}{2}}=3\)
\(3^3=27\)
\(\dfrac{125}{27}\)
Answer: 12527
The formula is.\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^m\)
\(27^{\frac{4}{3}}=\left(27^{\frac{1}{3}}\right)^{4}\)
\(27^{\frac{1}{3}}=3\)
\(3^4=81\)
Answer: 81
The formula is.\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^m\)
\(32^{\frac{4}{5}}=\left(32^{\frac{1}{5}}\right)^{4}\)
\(32^{\frac{1}{5}}=2\)
\(2^4=16\)
Answer: 16
The formula is.\(\left(\dfrac{a}{b}\right)^{\frac{m}{n}}=\dfrac{a^{\frac{m}{n}}}{b^{\frac{m}{n}}}\)
\(\left(\dfrac{8}{27}\right)^{\frac{2}{3}}=\dfrac{8^{\frac{2}{3}}}{27^{\frac{2}{3}}}\)
\(8^{\frac{2}{3}}=\left(8^{\frac{1}{3}}\right)^2\)
\(8^{\frac{1}{3}}=2\)
\(2^2=4\)
\(27^{\frac{2}{3}}=\left(27^{\frac{1}{3}}\right)^2\)
\(27^{\frac{1}{3}}=3\)
\(3^2=9\)
\(\dfrac{4}{9}\)
Answer: 49
The formula is.\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^m\)
\(81^{\frac{5}{4}}=\left(81^{\frac{1}{4}}\right)^{5}\)
\(81^{\frac{1}{4}}=3\)
\(3^5=243\)
Answer: 243
\(\left(\dfrac{27}{8}\right)^{\frac{2}{3}}=\dfrac{27^{\frac{2}{3}}}{8^{\frac{2}{3}}}\)
\(27^{\frac{2}{3}}=9\)
\(8^{\frac{2}{3}}=4\)
\(\dfrac{9}{4}\)
Answer: 94
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