3.2 Properties of Exponents

Before we start our discussion of exponential functions, we will introduce you to nine basic properties of exponents. It is useful to know the following properties when we deal with exponents and exponential functions. Let $m$ $m$ and $n$ $n$ be integers with $m > 0$ $m > 0$ and $n > 0$ $n > 0$ . Moreover, let $a$ $a$ and $b$ $b$ be positive real numbers, and let $u$ $u$ and $v$ $v$ be real numbers.

1. Definition of Exponents

The exponent of a number indicates how many times the number is multiplied by itself (used as a factor).

$a^{n} = a \cdot a \cdot a \cdot . . . \cdot a (a multiplied n times)$ $a^{n} = a \cdot a \cdot a \cdot . . . \cdot a (a multiplied n times)$

We call $a$ $a$ the base and $n$ $n$ the exponent.

2. Power of One

Any power of the integer 1 is 1.

$1^{u} = 1$ $1^{u} = 1$

3. Raised to the Power of Zero

Any nonzero number raised to the power of 0 equals 1.

$a^{0} = 1$ $a^{0} = 1$

4. Fractional Exponents

When the exponent of a number is a fraction, it is known as a fractional exponent. This notation expresses powers and roots together. The numerator of the fractional exponent is the power to which the number inside the root is raised, and the denominator of the fractional exponent is the root index.

$a^{\frac{m}{n}} = \sqrt[n]{a^{m}}$ $a^{\frac{m}{n}} = \sqrt[n]{a^{m}}$

Here are a few examples.

$5^{\frac{2}{3}} = \sqrt[3]{5^{2}} = \sqrt[3]{25}$ $5^{\frac{2}{3}} = \sqrt[3]{5^{2}} = \sqrt[3]{25}$

$10^{\frac{1}{5}} = \sqrt[5]{10}$ $10^{\frac{1}{5}} = \sqrt[5]{10}$

$2^{\frac{1}{2}} = \sqrt{2}$ $2^{\frac{1}{2}} = \sqrt{2}$

5. Negative Exponents

Negative exponents express the number of times 1 is divided by the base number.

$a^{- u} = {(\frac{1}{a})}^{u} = \frac{1}{a^{u}}$ $a^{- u} = {(\frac{1}{a})}^{u} = \frac{1}{a^{u}}$

Here are some examples:

$2^{- 2} = {(\frac{1}{2})}^{2} = \frac{1}{2^{2}}$ $2^{- 2} = {(\frac{1}{2})}^{2} = \frac{1}{2^{2}}$

$5^{- \frac{3}{2}} = {(\frac{1}{5})}^{\frac{3}{2}} = \frac{1}{5^{\frac{3}{2}}} = \frac{1}{\sqrt{5^{3}}}$ $5^{- \frac{3}{2}} = {(\frac{1}{5})}^{\frac{3}{2}} = \frac{1}{5^{\frac{3}{2}}} = \frac{1}{\sqrt{5^{3}}}$

$2 \cdot 3^{- 5} = 2 \cdot {(\frac{1}{3})}^{5} = \frac{2}{3^{5}}$ $2 \cdot 3^{- 5} = 2 \cdot {(\frac{1}{3})}^{5} = \frac{2}{3^{5}}$

6. Product of the Same Base

The product of the same number with different exponents is the number raised to the sum of all exponents.

$a^{u} a^{v} = a^{u + v}$ $a^{u} a^{v} = a^{u + v}$

For example,

$3^{2} 3^{7} = 3^{2 + 7} = 3^{9}$ $3^{2} 3^{7} = 3^{2 + 7} = 3^{9}$

$2^{4} 2^{- 1} = 2^{4 - 1} = 2^{3}$ $2^{4} 2^{- 1} = 2^{4 - 1} = 2^{3}$

$5^{\frac{1}{2}} 5^{\frac{2}{3}} = 5^{\frac{1}{2} + \frac{2}{3}} = 5^{\frac{7}{6}}$ $5^{\frac{1}{2}} 5^{\frac{2}{3}} = 5^{\frac{1}{2} + \frac{2}{3}} = 5^{\frac{7}{6}}$

7. Exponent Raised to Exponent

When a real number raised to an exponent is raised to another exponent, it is equal to the real number raised to the product of all exponents.

$(a^{u})^{v} = a^{u v}$ $(a^{u})^{v} = a^{u v}$

Following are some examples:

$(3^{2})^{4} = 3^{2 \cdot 4} = 3^{8}$ $(3^{2})^{4} = 3^{2 \cdot 4} = 3^{8}$

$(9^{3})^{- 2} = 3^{3 \cdot (- 2)} = 3^{- 6}$ $(9^{3})^{- 2} = 3^{3 \cdot (- 2)} = 3^{- 6}$

$(5^{3})^{\frac{4}{3}} = 5^{3 \cdot \frac{4}{3}} = 5^{4}$ $(5^{3})^{\frac{4}{3}} = 5^{3 \cdot \frac{4}{3}} = 5^{4}$

8. Power of a Product

When the product of two positive integers is raised to an exponent, it is equal to one positive integer raised to the exponent multiplied by the other positive integer raised to the exponent.

$(a b)^{u} = a^{u} b^{u}$ $(a b)^{u} = a^{u} b^{u}$

Here are some examples:

$(2 \cdot 3)^{4} = 2^{4} 3^{4}$ $(2 \cdot 3)^{4} = 2^{4} 3^{4}$

$(5 \cdot 2)^{- 3} = 5^{- 3} 2^{- 3}$ $(5 \cdot 2)^{- 3} = 5^{- 3} 2^{- 3}$

$(2^{- 2} 6^{\frac{1}{3}})^{4} = 2^{- 8} 6^{\frac{4}{3}}$ $(2^{- 2} 6^{\frac{1}{3}})^{4} = 2^{- 8} 6^{\frac{4}{3}}$

9. Exponents in Equations

If the bases of exponential numbers on both sides of an equation are the same, then the exponents of both sides of the equation are the same.

$If a^{u} = a^{v}, then u = v$ $If a^{u} = a^{v}, then u = v$ .

For example, if $2^{x} = 2^{3}$ $2^{x} = 2^{3}$ , then $x = 3$ $x = 3$ .

One more example. If $3^{x} = 9^{2}$ $3^{x} = 9^{2}$ , then $3^{x} = 9^{2} = (3^{2})^{2} = 3^{4}$ $3^{x} = 9^{2} = (3^{2})^{2} = 3^{4}$ . So, $x = 4$ $x = 4$ .

How about the following system of equations?

$\begin{matrix} 2^{x} = 4^{y + 2} 3^{9 y} = 27^{3 x + 9} \end{matrix}$ $\begin{aligned} 2^{x} = 4^{y + 2} \\ 3^{9 y} = 27^{3 x + 9} \end{aligned}$

Then we have

$\begin{matrix} 2^{x} & = 4^{y + 2} = (2^{2})^{y + 2} = 2^{2 y + 4} 3^{9 y} & = 27^{x + 3} = (3^{3})^{x + 3} = 3^{3 x + 9} \end{matrix}$ $\begin{aligned} 2^{x} & = 4^{y + 2} \\ = (2^{2})^{y + 2} \\ = 2^{2 y + 4} \\ 3^{9 y} & = 27^{x + 3} \\ = (3^{3})^{x + 3} \\ = 3^{3 x + 9} \end{aligned}$

So, we have

$\begin{matrix} x = 2 y + 4 9 y = 3 x + 9 \end{matrix}$ $\begin{aligned} x = 2 y + 4 \\ 9 y = 3 x + 9 \end{aligned}$

Solving the system of equations, we get $x = 18$ $x = 18$ and $y = 7$ $y = 7$ .

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