Chapter 1.2, Problem 35E

(a)

To determine

To simplify: given expression $x^8{x}^2$ .

Answer to Problem 35E

  $x^8{x}^2=x^{10}$

Explanation of Solution

Given information:

Given expression

  $x^8{x}^2$

Calculation:

Consider the given expression.

  $x^8{x}^2$

To simplify this, we will use the first law of exponents, which states

  $\begin{array}{c}{a}^m\cdot a^n=a^{m+n}\\ x^8{x}^2=x^{10}\end{array}$

(b)

To determine

To evaluate: the given expression $\left(3y^2\right)\left(4y^5\right)$ .

Answer to Problem 35E

  $\left(3y^2\right)\left(4y^5\right)=12y^7$

Explanation of Solution

Given information:

Given expression

  $\left(3y^2\right)\left(4y^5\right)$

Calculation:

Consider the given expression.

  $\left(3y^2\right)\left(4y^5\right)$

These four terms are multiplied, so we can write this as

  $\begin{array}{l}3\cdot 4\cdot y^2{y}^5\\ 12\cdot y^2{y}^5\end{array}$

To simplify this, we will use the first law of exponents, which states

  $\begin{array}{c}{a}^m\cdot a^n=a^{m+n}\\ 12\cdot y^2{y}^5=12\cdot y^{2+5}\\ 12\cdot y^2{y}^5=12\cdot y^7\\ \left(3y^2\right)\left(4y^5\right)=12y^7\end{array}$

(c)

To determine

To evaluate: given expression $x^2{x}^{-6}.$ .

Answer to Problem 35E

  $x^2{x}^{-6}=\frac{1}{x^4}$

Explanation of Solution

Given information:

Given expression

  $x^2{x}^{-6}.$

Calculation:

Consider the given expression.

  $x^2{x}^{-6}.$

To simplify this, we will use the first law of exponents, which states

  $\begin{array}{c}{a}^m\cdot a^n=a^{m+n}\\ x^2{x}^{-6}=x^{2+\left(-6\right)}\\ x^2{x}^{-6}=x^{-4}\end{array}$

Use the rule of negative exponents, which states

  $\begin{array}{l}{a}^{-n}=\frac{1}{a^n}\\ x^{-4}=\frac{1}{x^4}\\ x^2{x}^{-6}=\frac{1}{x^4}\end{array}$