Algebra Toolkit

Recall that the numerator of the exponent is the power and the denominator is the root. A negative exponent indicates a reciprocal.

1. \(\displaystyle x^{2/3} = \sqrt[3]{x^2}\)
2. \(\displaystyle t^{-3/2} = \dfrac{1}{t^{3/2}} = \dfrac{1}{\sqrt{t^3}}\)
3. \(\displaystyle w^{1.25} = w^{5/4} = \sqrt[4]{w^5}\)
4. \(\displaystyle z^{-0.4} = \dfrac{1}{z^{4/10}} = \dfrac{1}{z^{2/5}} = \dfrac{1}{\sqrt[5]{z^2}}\)

1. \(\displaystyle \sqrt[4]{b^3} = b^{3/4}\)
2. \(\displaystyle \dfrac{1}{\sqrt[6]{a^3}} = \dfrac{1}{a^{3/6}} = a^{-3/6} = a^{-1/2}\)
3. \(\displaystyle x^2\sqrt[4]{x} = x^2x^{1/4} = a^{2+1/4} = a^{9/4}\)
4. \(\displaystyle \dfrac{\sqrt[3]{v}}{\sqrt{v}} = \dfrac{v^{1/3}}{v^{1/2}} = v^{1/3-1/2} = v^{-1/6}\)

1. We can write both sides with base 3.

   \begin{equation*} ~~ 81 = 3^4,~~~\text{so} ~~x=4\text{.} \end{equation*}

2. We can write both sides with base 5.

   \begin{equation*} ~~125 = 5^3,~~~ \text{so}~~~5^{-3} = \dfrac{1}{125},~~~ \text{and} ~~x=-3\text{.} \end{equation*}
3. \(\left(\dfrac{3}{4}\right)^2 = \dfrac{9}{16}~\text{,}\) so \(~\left(\dfrac{3}{4}\right)^{-2} = \dfrac{16}{9}~~\text{,}\) and \(~x=-2\text{.}\)

4. We can write both sides with base 4.

   \begin{equation*} 64 = 4^3~~~ \text{and}~~~16 = 4^2\text{,} \end{equation*}

   so

   \begin{align*} ~(4^3)^x \amp = 4^2 \amp\amp \blert{\text{Multiply exponents.}}\\ 4^{3x} \amp = 4^2 \amp\amp \blert{\text{Equate exponents.}}\\ 3x \amp = 2\\ x \amp =\dfrac{2}{3} \end{align*}

1. \(2^4=16~\text{,}\) so we try a slightly smaller exponent and find that \(2^{3.9} = 14.9285~\text{,}\) so \(x \approx 3.9\text{.}\)
2. 65 is between \(3^3=27\) and \(3^4=81\text{,}\) so \(x\) must be between 3 and 4. By trying exponents 3.1, 3.2, 3.3, and so on, we find that \(3^{3.8}=65.022\text{,}\) so \(x \approx 3.8\text{.}\)
3. \(10^{-1}=0.1\) and \(10^{-2}=0.01\text{,}\) so \(-2 \lt x \lt -1\text{.}\) By trying exponents between \(-2\) and \(-1\text{,}\) we find that \(10^{-1.5} = 0.0316\text{,}\) so \(x \approx -1.5\text{.}\)
4. \(0.5^2 = 0.25\text{,}\) and as we increase the exponent on \(0.5\text{,}\) the result will be smaller. By trial and error we find that \(0.5^{2.3} = 0.2031\text{,}\) so \(x \approx 2.3\text{.}\)

When multiplying two powers with the same base, we add the exponents. Notice that the base does not change.

1. \(\displaystyle 1.35^6(1.35^4) = 1.35^{6+4}=1.35^{10}\)
2. \(\displaystyle 0.64^5(0.64^n) = 0.64^{5+n}\)

When dividing two powers with the same base, we subtract the exponents.

1. \(\displaystyle \dfrac{0.32^8}{0.32^2} = 0.32^{8-2} = 0.32^6\)
2. \(\displaystyle \dfrac{0.32^t}{0.32^x} = 0.32^{t-x}\)

When raising a power to a power, we multiply the exponents.

1. \(\displaystyle \left(1.07^5\right)^3 = 1.07^{15}\)
2. \(\displaystyle \left(1.07^4\right)^p = 1.07^{4p}\)