Chapter 7.2, Problem 46PPE

(a).

To determine

To find: The four different ways to write the given expression as a product of two powers with the same base using positive exponents.

Answer to Problem 46PPE

The four different ways to write the given expression as a product of two powers with the same base are $y^3{y}^3$ , $y^0{y}^6$ , $y^2{y}^4$ and $y^1{y}^6$ .

Explanation of Solution

Given information:

The given expression is $y^6$ .

Calculation:

Formula for the property of addition of exponents for any positive integer $n$ and $m$ is:

  $a^{m+n}=a^m\cdot a^n$

The given expression can be written in the first way as:

  $y^6=y^{3+3}$

Apply the addition property of exponents raise to a power in the above expression.

  $\begin{array}{c}{y}^6=y^{3+3}\\ =y^3{y}^3\end{array}$

The given expression can be written in the second way as:

  $y^6=y^{2+4}$

Apply the addition property of exponents raise to a power in the above expression.

  $\begin{array}{c}{y}^6=y^{2+4}\\ =y^2{y}^4\end{array}$

The given expression can be written in the third way as:

  $y^6=y^{1+5}$

Apply the addition property of exponents raise to a power in the above expression.

  $\begin{array}{c}{y}^6=y^{1+5}\\ =y^1{y}^5\end{array}$

The given expression can be written in the fourth way as:

  $y^6=y^{0+6}$

Apply the addition property of exponents raise to a power in the above expression.

  $\begin{array}{c}{y}^6=y^{0+6}\\ =y^0{y}^6\end{array}$

Therefore, the four different ways to write the given expression as a product of two powers with the same base are $y^3{y}^3$ , $y^0{y}^6$ , $y^2{y}^4$ and $y^1{y}^6$ .

(b).

To determine

To find: The four different ways to write the given expression as a product of two powers with the same base using negative or zero exponents in each product.

Answer to Problem 46PPE

The four different ways to write the given expression as a product of two powers with the same base ding zero exponents are $y^3{y}^3$ , $y^0{y}^6$ , $y^2{y}^4$ and $y^6{y}^0$ and using negative exponents are $y^9{y}^{-3}$ , $y^{12}{y}^{-6}$ , $y^{14}{y}^{-8}$ and $y^{19}{y}^{-13}$ .

Explanation of Solution

Given information:

The given expression is $y^6$ .

Calculation:

Formula for the property of addition of exponents for any positive integer $n$ and $m$ is:

  $a^{m+n}=a^m\cdot a^n$

The given expression can be written in the first way as:

  $y^6=y^{3+3}$

Apply the addition property of exponents raise to a power in the above expression.

  $\begin{array}{c}{y}^6=y^{3+3}\\ =y^3{y}^3\end{array}$

The given expression can be written in the second way as:

  $y^6=y^{2+4}$

Apply the addition property of exponents raise to a power in the above expression.

  $\begin{array}{c}{y}^6=y^{2+4}\\ =y^2{y}^4\end{array}$

The given expression can be written in the third way as:

  $y^6=y^{6+0}$

Apply the addition property of exponents raise to a power in the above expression.

  $\begin{array}{c}{y}^6=y^{0+6}\\ =y^6{y}^0\end{array}$

The given expression can be written in the fourth way as:

  $y^6=y^{0+6}$

Apply the addition property of exponents raise to a power in the above expression.

  $\begin{array}{c}{y}^6=y^{0+6}\\ =y^0{y}^6\end{array}$

So, the four different ways to write the given expression as a product of two powers with the same base using zero exponents are $y^3{y}^3$ , $y^0{y}^6$ , $y^2{y}^4$ and $y^0{y}^6$ .

For negative exponents, the expression can be written as:

  $y^{a+\left(-b\right)}=y^a\cdot y^{-b}$

Express the given expression using the above formula is:

First way:

  $\begin{array}{c}{y}^6=y^{12-6}\\ =y^{12}{y}^{-6}\end{array}$

Second way:

  $\begin{array}{c}{y}^6=y^{9-3}\\ =y^9{y}^{-3}\end{array}$

Third way:

  $\begin{array}{c}{y}^6=y^{14-8}\\ =y^{14}{y}^{-8}\end{array}$

Fourth way:

  $\begin{array}{c}{y}^6=y^{19-13}\\ =y^{19}{y}^{-13}\end{array}$

So, the four different ways to write the given expression as a product of two powers with the same base using negative exponents are $y^9{y}^{-3}$ , $y^{12}{y}^{-6}$ , $y^{14}{y}^{-8}$ and $y^{19}{y}^{-13}$ .

Therefore, the four different ways to write the given expression as a product of two powers with the same base ding zero exponents are $y^3{y}^3$ , $y^0{y}^6$ , $y^2{y}^4$ and $y^6{y}^0$ and using negative exponents are $y^9{y}^{-3}$ , $y^{12}{y}^{-6}$ , $y^{14}{y}^{-8}$ and $y^{19}{y}^{-13}$ .

(c).

To determine

To find: The reason and different number of ways to write the given expression as a product of two powers.

Explanation of Solution

Given information:

The given expression is $y^6$ .

Reason:

As calculated in part (a) and (b), the given expression can be written as positive, negative and zero exponents in four different ways each. However, the exponents can be infinitely large in numbers for negative exponents as the power $6$ can be obtained from the difference of two numbers. Thus, there are an infinite number of ways to write the given expression.