# Exponents

Rules of Exponents

Exponent problems like the ones below can usually be solved by manipulating the component parts
according to the rules of exponents, which we summarize as:

1. xa ∙ xb = xa+b Example: 23 ∙ 24 = 27, or 8 ∙ 16 = 128 = 27
2. (xa)b = xab Example: (32)3 = 323 = 36, or 93 = 729, and 36 = 729
3. xa / xb = xa-b Example: 24/23 = 24-3 = 21, or 16/8 = 2
4. x-a = 1/xa Example: 2-3 = 1/23 = 1/8
5. 1/x-a = xa Example: 1/3-2 = 32 = 9
6. \begin{align}x^\frac{m}{n} = \sqrt[n]{x^m}\end{align} Example: \begin{align}2^\frac{2}{3} = \sqrt[3]{2^2} = \sqrt[3]{4}\end{align}
7. xa ∙ ya = (xy)a Example: 3² · 4² = (3 · 4)² = 12² = 144 = 9 · 16 = 3² · 4²

Example 1)

\begin{align}10^{-2} + (10^{-2})^2 + (10^{-2})^3 =\end{align}

A. .010010001
B. .010101
C. 101.01
D. 121.001
E. .010203

Explanation:

\begin{align}10^{-2} + (10^{-2})^2 + (10^{-2})^3 =\end{align}
\begin{align}10^{-2} + (10^{-4}) + (10^{-6}) =\end{align}
.01 + .0001 + .000001 = .010101

So B is the correct answer.

Example 2)

\begin{align}5^{-6} =\end{align}

I. \begin{align}6.4 \times 10^{-5}\end{align}
II. \begin{align}(.2)^6\end{align}
III. .000064

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III

Explanation: The possible answers listed for this problem look surprising, because none of them seem to have anything to do with 5. But notice how they do involve 2 and 10, and 10 = 2 ∙ 5. So that should give us a hint that we need to do something involving 2, 5, and 10. First of all, note that 5 = 10/2, so

\begin{align}5^{-6} = (\frac{10}{2})^{-6} = (\frac{10^{-6}}{2^{-6}})\end{align}

But since negative exponents turn positive when moved from the numerator to denominator, and vice versa, we can turn this fraction upside down and change the negative exponents to positives:

\begin{align}(\frac{10^{-6}}{2^{-6}}) = (\frac{2^{6}}{10^{6}}) = 64 \times 10^{-6} = 6.4 \times 10^{-5}\end{align}

So answer choice I is correct.

Notice also that \begin{align}(\frac{2^{6}}{10^{6}}) = (\frac{2}{10})^6 = (.2)^6\end{align}, so answer choice II is correct also.

Finally, \begin{align}6.4 \times 10^{-5} = .000064\end{align}, so answer choice III is correct.

Therefore, the correct final answer to the question is E.

Example 3)

\begin{align}(\frac{2}{3})^{-3} \cdot (\frac{3}{2})^{-4} \cdot (2)^{-1} = \end{align}

A. \begin{align}(\frac{3}{2})^7 \cdot 2\end{align}
B. 1/3
C. 2/3
D. \begin{align}(\frac{6}{9})^{-7}\end{align}
E. 4/3

Explanation:

\begin{align}(\frac{2}{3})^{-3} \cdot (\frac{3}{2})^{-4} \cdot (2)^{-1} = \end{align}

\begin{align}(\frac{3}{2})^3 \cdot (\frac{2}{3})^4 \cdot \frac{1}{2} = \end{align}

\begin{align}(\frac{3^{3} \cdot 2^{4}}{2^{3} \cdot 3^{4}}) \cdot \frac{1}{2} = \end{align}

\begin{align}\frac{2}{3} \cdot \frac{1}{2} = \end{align}

\begin{align}\frac{1}{3}\end{align}

So B is the correct answer.