# **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:

- x
^{a}∙ x^{b}= x^{a+b}Example: 2^{3 }∙ 2^{4}= 2^{7}, or 8 ∙ 16 = 128 = 2^{7} - (x
^{a})^{b}= x^{ab}Example: (3^{2})^{3}= 3^{2}^{∙}^{3}= 3^{6}, or 9^{3}= 729, and 3^{6}= 729 - x
^{a}/ x^{b}= x^{a-b}Example: 2^{4}/2^{3}= 2^{4-3}= 2^{1}, or 16/8 = 2 - x
^{-a}= 1/x^{a}Example: 2^{-3}= 1/2^{3}= 1/8 - 1/x
^{-a}= x^{a}Example: 1/3^{-2}= 3^{2}= 9 - \(\begin{align}x^\frac{m}{n} = \sqrt[n]{x^m}\end{align}\) Example: \(\begin{align}2^\frac{2}{3} = \sqrt[3]{2^2} = \sqrt[3]{4}\end{align}\)
- x
^{a}∙ y^{a}= (xy)^{a}Example: 3² · 4² = (3 · 4)² = 12² = 144 = 9 · 16 = 3² · 4²

On this page we give 3 examples.

**To practice more of these types of problems, click** here.

**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.**

**To practice more of these types of problems, click** here.