IB AA ยท 12A Laws of Exponents
Raise both sides to the reciprocal exponent, then evaluate. If two powers of x appear, divide so one power remains.
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Raise to the reciprocal
If the exponent is a fraction, raise both sides to the reciprocal of that fraction.
\(x^{\frac{3}{2}}=8\)
\(x=8^{\frac{2}{3}}\)
\(4\)
Then evaluate
After you raise both sides, evaluate the rational exponent the same way as on Evaluate Rational Exponents: take the root first.
A negative exponent
Raising to the reciprocal still works, and the reciprocal of a negative is negative.
\(x^{-\frac{1}{2}}=\dfrac{1}{4}\)
\(x=\left(\dfrac{1}{4}\right)^{-2}\)
16
Same variable, two roots
Write both sides as powers of x. Divide so one power remains. Assume x > 0 when an even root is involved.
Raise both sides to the reciprocal. Take the root first.
\(x^{\frac{3}{2}}=8\)
Raise both sides to the reciprocal exponent.\(\left(x^{\frac{3}{2}}\right)^{\frac{2}{3}}=x\)
\(x=8^{\frac{2}{3}}\)
\(8^{\frac{2}{3}}=\left(8^{\frac{1}{3}}\right)^2\)
\(8^{\frac{1}{3}}=2\)
\(2^2=4\)
Answer: x = 4
\(x=8^{\frac{4}{3}}\)
\(8^{\frac{4}{3}}=\left(8^{\frac{1}{3}}\right)^4\)
\(8^{\frac{1}{3}}=2\)
\(2^4=16\)
Answer: x = 16
\(x=4^{\frac{5}{2}}\)
\(4^{\frac{5}{2}}=\left(4^{\frac{1}{2}}\right)^5\)
\(4^{\frac{1}{2}}=2\)
\(2^5=32\)
Answer: x = 32
\(x=32^{\frac{2}{5}}\)
\(32^{\frac{2}{5}}=\left(32^{\frac{1}{5}}\right)^2\)
\(32^{\frac{1}{5}}=2\)
\(2^2=4\)
Answer: x = 4
\(x=16^{\frac{3}{4}}\)
\(16^{\frac{3}{4}}=\left(16^{\frac{1}{4}}\right)^3\)
\(16^{\frac{1}{4}}=2\)
\(2^3=8\)
Answer: x = 8
\(x=8^{\frac{5}{3}}\)
\(8^{\frac{5}{3}}=\left(8^{\frac{1}{3}}\right)^5\)
\(8^{\frac{1}{3}}=2\)
\(2^5=32\)
Answer: x = 32
A negative exponent or a unit fraction on the right. Flip, then evaluate.
\(x^{-\frac{1}{2}}=\dfrac{1}{4}\)
Raise both sides to the reciprocal exponent.\(\left(x^{-\frac{1}{2}}\right)^{-2}=x\)
\(x=\left(\dfrac{1}{4}\right)^{-2}\)
The formula is.\(\left(\dfrac{a}{b}\right)^{-n}=\left(\dfrac{b}{a}\right)^{n}\)
\(\left(\dfrac{4}{1}\right)^{2}\)
\(4^2=16\)
Answer: x = 16
\(x=\left(\dfrac{1}{2}\right)^{-3}\)
\(2^3=8\)
Answer: x = 8
\(x=\left(\dfrac{1}{8}\right)^{-\frac{2}{3}}\)
\(8^{\frac{2}{3}}=\left(8^{\frac{1}{3}}\right)^2\)
\(8^{\frac{1}{3}}=2\)
\(2^2=4\)
Answer: x = 4
\(x=\left(\dfrac{1}{9}\right)^{-\frac{3}{2}}\)
\(9^{\frac{3}{2}}=\left(9^{\frac{1}{2}}\right)^3\)
\(9^{\frac{1}{2}}=3\)
\(3^3=27\)
Answer: x = 27
\(x=\left(\dfrac{1}{8}\right)^{\frac{2}{3}}\)
\(\dfrac{1^{\frac{2}{3}}}{8^{\frac{2}{3}}}=\dfrac{1}{4}\)
Answer: \(x=\dfrac{1}{4}\)
\(x=\left(\dfrac{3}{2}\right)^{-2}\)
\(\left(\dfrac{2}{3}\right)^{2}=\dfrac{4}{9}\)
Answer: \(x=\dfrac{4}{9}\)
Two powers of x on the same side, or a reciprocal that is not a unit fraction.
\(x^{\frac{1}{2}}=3x^{\frac{1}{3}}\)
Divide both sides by \(x^{\frac{1}{3}}\).\(\dfrac{a^m}{a^n}=a^{m-n}\)
\(x^{\frac{1}{2}-\frac{1}{3}}=3\)
\(\dfrac{1}{2}=\dfrac{3}{6}\)
\(\dfrac{1}{3}=\dfrac{2}{6}\)
\(\dfrac{3}{6}-\dfrac{2}{6}=\dfrac{1}{6}\)
\(x^{\frac{1}{6}}=3\)
\(x=3^6\)
\(3^6=729\)
Answer: x = 729
\(x^{\frac{1}{2}}=4x^{\frac{1}{3}}\)
\(x^{\frac{1}{2}-\frac{1}{3}}=4\)
\(x^{\frac{1}{6}}=4\)
\(x=4^6\)
\(4^2=16\)
\(4^4=256\)
\(4^6=256\cdot 16\)
\(4096\)
Answer: x = 4096
\(x=27^{\frac{2}{3}}\)
\(27^{\frac{2}{3}}=\left(27^{\frac{1}{3}}\right)^2\)
\(27^{\frac{1}{3}}=3\)
\(3^2=9\)
Answer: x = 9
\(x^{\frac{1}{3}}=2x^{\frac{1}{2}}\)
\(1=2x^{\frac{1}{2}-\frac{1}{3}}\)
\(1=2x^{\frac{1}{6}}\)
\(x^{\frac{1}{6}}=\dfrac{1}{2}\)
\(x=\left(\dfrac{1}{2}\right)^6\)
\(\dfrac{1}{64}\)
Answer: \(x=\dfrac{1}{64}\)
\(x=\left(\dfrac{1}{8}\right)^{-\frac{4}{3}}\)
\(8^{\frac{4}{3}}=\left(8^{\frac{1}{3}}\right)^4\)
\(8^{\frac{1}{3}}=2\)
\(2^4=16\)
Answer: x = 16
\(x=\left(\dfrac{1}{32}\right)^{\frac{2}{5}}\)
\(32^{\frac{2}{5}}=\left(32^{\frac{1}{5}}\right)^2\)
\(32^{\frac{1}{5}}=2\)
\(2^2=4\)
\(\dfrac{1}{4}\)
Answer: \(x=\dfrac{1}{4}\)
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