Q. 108 - Intermediate Algebra

Question

Use the definition of Negative Exponent to simplify (a)$3·4^{-2}$ (b)${\left(3·4\right)}^{-2}$

Step-by-Step Solution

Answer

The simplified equation is (a) $\frac{3}{16}$ (b) $\frac{1}{144}.$

Part (a). Step 1. Given information.

The given expression is $3·4^{-2}$.

We have to simplify the equation.

Part (a). Step 2. Definition of Negative exponents.

If $n$ is an integer and $a\ne 0$ then

$a^{-n}=\frac{1}{a^n}$.

Part (a). Step 3. Simplify the expression.

Using the definition of Negative exponents

$$\begin{gathered} 3·4^{-2}=3·{\left(\frac{1}{4}\right)}^2 \\ =3·\left(\frac{1}{16}\right) \\ =\frac{3}{16}. \end{gathered}$$

So, $3·4^{-2}=\frac{3}{16}.$

Part (b). Step 1. Given information.

The given expression is ${\left(3·4\right)}^{-2}.$

We have to simplify the expression.

Part (b). Step 2. Definition of Negative exponents.

If $n$ is an integer and $a\ne 0$ then

$a^{-n}=\frac{1}{a^n}.$

Part (b). Step 3. Simplify the expression.

Using the definition of Negative exponents

$$\begin{gathered} {\left(3·4\right)}^{-2}={\left(\frac{1}{3}·\frac{1}{4}\right)}^2 \\ =\frac{1}{{12}^2} \\ =\frac{1}{144} \end{gathered}$$.

So, ${\left(3·4\right)}^{-2}=\frac{1}{144}.$