# 2.4 Exponents  (Page 2/2)

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${\left(8x\right)}^{3}$ means $\left(8x\right)\left(8x\right)\left(8x\right)$ since the parentheses indicate that the exponent 3 is directly connected to the factor $8x$ . Remember that the grouping symbols indicate that the quantities inside are to be considered as one single number.

$34{\left(a+1\right)}^{2}$ means $34\cdot \left(a+1\right)\left(a+1\right)$ since the exponent 2 applies only to the factor $\left(a+1\right)$ .

## Practice set b

Write each of the following without exponents.

$4{a}^{3}$

$4aaa$

${\left(4a\right)}^{3}$

$\left(4a\right)\left(4a\right)\left(4a\right)$

## Sample set c

Select a number to show that ${\left(2x\right)}^{2}$ is not always equal to $2{x}^{2}$ .

Suppose we choose $x$ to be 5. Consider both ${\left(2x\right)}^{2}$ and $2{x}^{2}$ .

$\begin{array}{lll}{\left(2x\right)}^{2}\hfill & \hfill & 2{x}^{2}\hfill \\ {\left(2\cdot 5\right)}^{2}\hfill & \hfill & 2\cdot {5}^{2}\hfill \\ {\left(10\right)}^{2}\hfill & \hfill & 2\cdot 25\hfill \\ 100\hfill & \ne \hfill & 50\hfill \end{array}$

Notice that ${\left(2x\right)}^{2}=2{x}^{2}$ only when $x=0$ .

## Practice set c

Select a number to show that ${\left(5x\right)}^{2}$ is not always equal to $5{x}^{2}$ .

Select $x=3$ . Then ${\left(5\cdot 3\right)}^{2}={\left(15\right)}^{2}=225$ , but $5\cdot {3}^{2}=5\cdot 9=45$ .     $225\ne 45$ .

In ${x}^{n}$ ,

## Base

$x$ is the base

## Exponent

$n$ is the exponent

## Power

The number represented by ${x}^{n}$ is called a power .

## $x$ To the $n$ Th power

The term ${x}^{n}$ is read as " $x$ to the $n$ th power," or more simply as " $x$ to the $n$ th."

## $x$ Squared and $x$ Cubed

The symbol ${x}^{2}$ is often read as " $x$ squared," and ${x}^{3}$ is often read as " $x$ cubed." A natural question is "Why are geometric terms appearing in the exponent expression?" The answer for ${x}^{3}$ is this: ${x}^{3}$ means $x\cdot x\cdot x$ . In geometry, the volume of a rectangular box is found by multiplying the length by the width by the depth. A cube has the same length on each side. If we represent this length by the letter $x$ then the volume of the cube is $x\cdot x\cdot x$ , which, of course, is described by ${x}^{3}$ . (Can you think of why ${x}^{2}$ is read as $x$ squared?)

Cube with
length $=x$
width $=x$
depth $=x$
Volume $=xxx={x}^{3}$

## The order of operations

In Section [link] we were introduced to the order of operations. It was noted that we would insert another operation before multiplication and division. We can do that now.

## The order of operations

1. Perform all operations inside grouping symbols beginning with the innermost set.
2. Perform all exponential operations as you come to them, moving left-to-right.
3. Perform all multiplications and divisions as you come to them, moving left-to-right.
4. Perform all additions and subtractions as you come to them, moving left-to-right.

## Sample set d

Use the order of operations to simplify each of the following.

${2}^{2}+5=4+5=9$

${5}^{2}+{3}^{2}+10=25+9+10=44$

$\begin{array}{ll}{2}^{2}+\left(5\right)\left(8\right)-1\hfill & =4+\left(5\right)\left(8\right)-1\hfill \\ \hfill & =4+40-1\hfill \\ \hfill & =43\hfill \end{array}$

$\begin{array}{ll}7\cdot 6-{4}^{2}+{1}^{5}\hfill & =7\cdot 6-16+1\hfill \\ \hfill & =42-16+1\hfill \\ \hfill & =27\hfill \end{array}$

$\begin{array}{ll}{\left(2+3\right)}^{3}+{7}^{2}-3{\left(4+1\right)}^{2}\hfill & ={\left(5\right)}^{3}+{7}^{2}-3{\left(5\right)}^{2}\hfill \\ \hfill & =125+49-3\left(25\right)\hfill \\ \hfill & =125+49-75\hfill \\ \hfill & =99\hfill \end{array}$

$\begin{array}{ll}{\left[4{\left(6+2\right)}^{3}\right]}^{2}\hfill & ={\left[4{\left(8\right)}^{3}\right]}^{2}\hfill \\ \hfill & ={\left[4\left(512\right)\right]}^{2}\hfill \\ \hfill & ={\left[2048\right]}^{2}\hfill \\ \hfill & =4,194,304\hfill \end{array}$

$\begin{array}{ll}6\left({3}^{2}+{2}^{2}\right)+{4}^{2}\hfill & =6\left(9+4\right)+{4}^{2}\hfill \\ \hfill & =6\left(13\right)+{4}^{2}\hfill \\ \hfill & =6\left(13\right)+16\hfill \\ \hfill & =78+16\hfill \\ \hfill & =94\hfill \end{array}$

$\begin{array}{ll}\frac{{6}^{2}+{2}^{2}}{{4}^{2}+6\cdot {2}^{2}}+\frac{{1}^{3}+{8}^{2}}{{10}^{2}-\left(19\right)\left(5\right)}\hfill & =\frac{36+4}{16+6\cdot 4}+\frac{1+64}{100-95}\hfill \\ \hfill & =\frac{36+4}{16+24}+\frac{1+64}{100-95}\hfill \\ \hfill & =\frac{40}{40}+\frac{65}{5}\hfill \\ \hfill & =1+13\hfill \\ \hfill & =14\hfill \end{array}$

## Practice set d

Use the order of operations to simplify the following.

${3}^{2}+4\cdot 5$

29

${2}^{3}+{3}^{3}-8\cdot 4$

3

${1}^{4}+{\left({2}^{2}+4\right)}^{2}÷{2}^{3}$

9

${\left[6\left(10-{2}^{3}\right)\right]}^{2}-{10}^{2}-{6}^{2}$

8

$\frac{{5}^{2}+{6}^{2}-10}{1+{4}^{2}}+\frac{{0}^{4}-{0}^{5}}{{7}^{2}-6\cdot {2}^{3}}$

3

## Exercises

For the following problems, write each of the quantities using exponential notation.

$b$ to the fourth

${b}^{4}$

$a$ squared

$x$ to the eighth

${x}^{8}$

$\left(-3\right)$ cubed

5 times $s$ squared

$5{s}^{2}$

3 squared times $y$ to the fifth

$a$ cubed minus $\left(b+7\right)$ squared

${a}^{3}-{\left(b+7\right)}^{2}$

$\left(21-x\right)$ cubed plus $\left(x+5\right)$ to the seventh

$xxxxx$

${x}^{5}$

$\left(8\right)\left(8\right)xxxx$

$2\cdot 3\cdot 3\cdot 3\cdot 3xxyyyyy$

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

$2\cdot 2\cdot 5\cdot 6\cdot 6\cdot 6xyyzzzwwww$

$7xx\left(a+8\right)\left(a+8\right)$

$7{x}^{2}{\left(a+8\right)}^{2}$

$10xyy\left(c+5\right)\left(c+5\right)\left(c+5\right)$

$4x4x4x4x4x$

${\left(4x\right)}^{5}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}{4}^{5}{x}^{5}$

$\left(9a\right)\left(9a\right)\left(9a\right)\left(9a\right)$

$\left(-7\right)\left(-7\right)\left(-7\right)aabbba\left(-7\right)baab$

${\left(-7\right)}^{4}{a}^{5}{b}^{5}$

$\left(a-10\right)\left(a-10\right)\left(a+10\right)$

$\left(z+w\right)\left(z+w\right)\left(z+w\right)\left(z-w\right)\left(z-w\right)$

${\left(z+w\right)}^{3}{\left(z-w\right)}^{2}$

$\left(2y\right)\left(2y\right)2y2y$

$3xyxxy-\left(x+1\right)\left(x+1\right)\left(x+1\right)$

$3{x}^{3}{y}^{2}-{\left(x+1\right)}^{3}$

For the following problems, expand the quantities so that no exponents appear.

${4}^{3}$

${6}^{2}$

$6\text{\hspace{0.17em}}·\text{\hspace{0.17em}}6$

${7}^{3}{y}^{2}$

$8{x}^{3}{y}^{2}$

$8\text{\hspace{0.17em}}·\text{\hspace{0.17em}}x\text{\hspace{0.17em}}·\text{\hspace{0.17em}}x\text{\hspace{0.17em}}·\text{\hspace{0.17em}}x\text{\hspace{0.17em}}·\text{\hspace{0.17em}}y\text{\hspace{0.17em}}·\text{\hspace{0.17em}}y$

${\left(18{x}^{2}{y}^{4}\right)}^{2}$

${\left(9{a}^{3}{b}^{2}\right)}^{3}$

$\left(9aaabb\right)\left(9aaabb\right)\left(9aaabb\right)\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}9\text{\hspace{0.17em}}·\text{\hspace{0.17em}}9\text{\hspace{0.17em}}·\text{\hspace{0.17em}}9aaaaaaaaabbbbbb$

$5{x}^{2}{\left(2{y}^{3}\right)}^{3}$

$10{a}^{3}{b}^{2}{\left(3c\right)}^{2}$

$10aaabb\left(3c\right)\left(3c\right)\text{\hspace{0.17em}or}\text{\hspace{0.17em}}10\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3aaabbcc$

${\left(a+10\right)}^{2}{\left({a}^{2}+10\right)}^{2}$

$\left({x}^{2}-{y}^{2}\right)\left({x}^{2}+{y}^{2}\right)$

$\left(xx-yy\right)\left(xx+yy\right)$

For the following problems, select a number (or numbers) to show that

${\left(5x\right)}^{2}$ is not generally equal to $5{x}^{2}$ .

${\left(7x\right)}^{2}$ is not generally equal to $7{x}^{2}$ .

Select $x=2.$ Then, $196\ne 28.$

${\left(a+b\right)}^{2}$ is not generally equal to ${a}^{2}+{b}^{2}$ .

For what real number is ${\left(6a\right)}^{2}$ equal to $6{a}^{2}$ ?

zero

For what real numbers, $a$ and $b$ , is ${\left(a+b\right)}^{2}$ equal to ${a}^{2}+{b}^{2}$ ?

Use the order of operations to simplify the quantities for the following problems.

${3}^{2}+7$

16

${4}^{3}-18$

${5}^{2}+2\left(40\right)$

105

${8}^{2}+3+5\left(2+7\right)$

${2}^{5}+3\left(8+1\right)$

59

${3}^{4}+{2}^{4}{\left(1+5\right)}^{3}$

$\left({6}^{2}-{4}^{2}\right)÷5$

4

${2}^{2}\left(10-{2}^{3}\right)$

$\left({3}^{4}-{4}^{3}\right)÷17$

1

${\left(4+3\right)}^{2}+1÷\left(2\cdot 5\right)$

${\left({2}^{4}+{2}^{5}-{2}^{3}\cdot 5\right)}^{2}÷{4}^{2}$

4

${1}^{6}+{0}^{8}+{5}^{2}{\left(2+8\right)}^{3}$

$\left(7\right)\left(16\right)-{9}^{2}+4\left({1}^{1}+{3}^{2}\right)$

71

$\frac{{2}^{3}-7}{{5}^{2}}$

$\frac{{\left(1+6\right)}^{2}+2}{19}$

$\frac{51}{19}$

$\frac{{6}^{2}-1}{5}+\frac{{4}^{3}+\left(2\right)\left(3\right)}{10}$

$\frac{5\left[{8}^{2}-9\left(6\right)\right]}{{2}^{5}-7}+\frac{{7}^{2}-{4}^{2}}{{2}^{4}-5}$

5

$\frac{{\left(2+1\right)}^{3}+{2}^{3}+{1}^{3}}{{6}^{2}}-\frac{{15}^{2}-{\left[2\left(5\right)\right]}^{2}}{5\cdot {5}^{2}}$

$\frac{{6}^{3}-2\cdot {10}^{2}}{{2}^{2}}+\frac{18\left({2}^{3}+{7}^{2}\right)}{2\left(19\right)-{3}^{3}}$

$\frac{1070}{11}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}97.\overline{27}$

## Exercises for review

( [link] ) Use algebraic notation to write the statement "a number divided by eight, plus five, is equal to ten."

( [link] ) Draw a number line that extends from $-5$ to 5 and place points at all real numbers that are strictly greater than $-3$ but less than or equal to 2.

( [link] ) Is every integer a whole number?

( [link] ) Use the commutative property of multiplication to write a number equal to the number $yx$ .

$xy$

( [link] ) Use the distributive property to expand $3\left(x+6\right)$ .

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