Chapter 7.3, Problem 73PPE

To determine

To find: The different number of ways to write the given expression using a property of raising a product to a power.

Answer to Problem 73PPE

The different number of ways to write the given expression using a property of raising a product to a power is $10$ .

Explanation of Solution

Given information: The given expression is $16x^4$ .

Calculation:

The formula for property of raising a product to a power for any positive integer $n$ is:

  ${\left(ab\right)}^n=a^n{b}^n$

The given expression can be written by using the property of raising a product to a power as:

First way:

  $\begin{array}{c}\left(16x^4\right)={\left(2\right)}^4\cdot x^4\\ ={\left(2x\right)}^4\end{array}$

Second way:

  $\begin{array}{c}16x^4={\left(2x\right)}^4\left[\therefore a^x=\frac{1}{a^{-x}}\right]\\ ={\left(\frac{1}{2x}\right)}^{-4}\end{array}$

Third way:

  $\begin{array}{c}16x^4={\left(2x\right)}^4\\ ={\left(-2x\right)}^4\end{array}$

Fourth way:

  $\begin{array}{c}16x^4={\left(-2x\right)}^4\\ ={\left(\frac{1}{-2x}\right)}^{-4}\end{array}$

Fifth way:

  $16x^4={\left(16x^4\right)}^1$

Sixth way:

  $\begin{array}{c}16x^4={\left(16x^4\right)}^1\left[\therefore a^x=\frac{1}{a^{-x}}\right]\\ ={\left(\frac{1}{16x^4}\right)}^{-1}\end{array}$

Seventh way: The factors of $16$ and $x^4$ are:

  $\begin{array}{c}16=4\times 4={\left(4\right)}^2\\ x^4=x^2\times x^2={\left(x^2\right)}^2\end{array}$

The product of $16$ and $x^4$ is:

  $16x^4={\left(4x^2\right)}^2$

Eighth way:

  $\begin{array}{c}16x^4={\left(4x^2\right)}^2\left[\therefore a^x=\frac{1}{a^{-x}}\right]\\ ={\left(\frac{1}{4x^2}\right)}^{-2}\end{array}$

Ninth way:

  $\begin{array}{c}16x^4={\left(\frac{1}{4x^2}\right)}^{-2}\\ ={\left(-4x^2\right)}^2\end{array}$

Tenth way:

  $\begin{array}{c}16x^4={\left(-4x^2\right)}^2\\ ={\left(\frac{1}{-4x^2}\right)}^2\end{array}$

Therefore, the different number of ways to write the given expression using a property of raising a product to a power is $10$ .