Chapter 5, Problem 1TE

In Exercises 1 and 2, use the properties of exponents to simplify.

1.

1. a. x⁴ · x³
2. b. $\frac{x^{12}}{x^7}$
3. c. (4ay)⁴
4. d. ${\left(\frac{3}{4}\right)}^4$
5. e. 2³ · 2²
6. f. (x⁴)²

To determine

To simplify: The expression $x^4\cdot x^3$ by using the properties of exponents.

Answer to Problem 1TE

The simplified form of $x^4\cdot x^3$ is $x^7$.

Explanation of Solution

Formula used:

Product property:

For real number a and integers m and n, the product property is $a^m\cdot a^n=a^{m+n}$.

Calculation:

The given expression is $x^4\cdot x^3$.

Here, $a=x$, $m=4$ and $n=3$.

Use product property and simplify the expression as follows.

$\begin{array}{c}{x}^4\cdot x^3=x^{4+3}\\ =x^7\end{array}$

Therefore, the simplified form of $x^4\cdot x^3$ is $x^7$.

To determine

To simplify: The expression $\frac{x^{12}}{x^7}$ by using the properties of exponents.

Answer to Problem 1TE

The simplified form of $\frac{x^{12}}{x^7}$ is $x^5$.

Explanation of Solution

Formula used:

Quotient property:

For real number a and integers m and n, the quotient property is $\frac{a^m}{a^n}=a^{m-n},a\ne 0$.

Calculation:

The given expression is $\frac{x^{12}}{x^7}$.

Here, $a=x$, $m=12$ and $n=7$.

Use quotient property and simplify the expression as follows.

$\begin{array}{c}\frac{x^{12}}{x^7}=x^{12-7}\\ =x^5\end{array}$

Therefore, the simplified form of $\frac{x^{12}}{x^7}$ is $x^5$.

To determine

To simplify: The expression ${\left(4ay\right)}^4$ by using the properties of exponents.

Answer to Problem 1TE

The simplified form of ${\left(4ay\right)}^4$ is $256a^4{y}^4$.

Explanation of Solution

Formula used:

Power of a product property:

For real numbers a and b and integers m and n, the power of product property is ${\left(ab\right)}^m=a^m{b}^m$.

Calculation:

The given expression is ${\left(4ay\right)}^4$.

Here, $a=4a$, $b=y$ and $m=4$.

Use the above property and simplify the expression as follows.

$\begin{array}{c}{\left(4ay\right)}^4={\left(4a\right)}^4{\left(y\right)}^4\\ =256a^4{y}^4\end{array}$

Therefore, the simplified form of ${\left(4ay\right)}^4$ is $256a^4{y}^4$.

To determine

To simplify: The expression ${\left(\frac{3}{z}\right)}^4$ by using the properties of exponents.

Answer to Problem 1TE

The simplified form of ${\left(\frac{3}{z}\right)}^4$ is $\frac{3^4}{z^4}$.

Explanation of Solution

Formula used:

Power of a Quotient property:

For real numbers a and b and integers m and n, the power of Quotient property is ${\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m},\text{ for }b\ne 0$.

Calculation:

The given expression is ${\left(\frac{3}{z}\right)}^4$.

Here, $a=3$, $b=z$ and $m=4$.

Use the above property and simplify the expression as follows.

$\begin{array}{c}{\left(\frac{3}{z}\right)}^4=\frac{3^4}{z^4}\\ =\frac{81}{z^4}\end{array}$

Therefore, the simplified form of ${\left(\frac{3}{z}\right)}^4$ is $\frac{81}{z^4}$.

To determine

To simplify: The expression $2^3\cdot 2^2$ by the using properties of exponents.

Answer to Problem 1TE

The simplified form of $2^3\cdot 2^2$ is 32.

Explanation of Solution

Formula used:

Product property:

For real number a and integers m and n, the product property is $a^m\cdot a^n=a^{m+n}$.

Calculation:

The given expression is $2^3\cdot 2^2$.

Here, $a=2$, $m=3$ and $n=2$.

Use the product property and simplify the expression as follows.

$\begin{array}{c}{2}^3\cdot 2^2=2^{3+2}\\ =2^5\\ =32\end{array}$

Therefore, the simplified form of $2^3\cdot 2^2$ is 32.

To determine

To simplify: The expression ${\left(x^4\right)}^2$ by using the properties of exponents.

Answer to Problem 1TE

The simplified form of ${\left(x^4\right)}^2$ is $x^8$.

Explanation of Solution

Formula used:

Power of a power property:

For real number a and integers m and n, the power of a power property is ${\left(a^m\right)}^n=a^{mn}$.

Calculation:

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

Note that here, $a=x$, $m=4\text{ and }n=2$.

Use the above property and simplify the expression as follows.

$\begin{array}{c}{\left(x^4\right)}^2=x^{4\cdot 2}\\ =x^8\end{array}$

Therefore, the simplified form of ${\left(x^4\right)}^2$ is $x^8$.