7.1 Basic properties of exponents (Page 2/2)

Page 2 / 2

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

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

Practice set b

Write each of the following without exponents.

$4 a^{3}$

$4 a a a$

${(4 a)}^{3}$

$(4 a) (4 a) (4 a)$

Sample set c

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

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

\begin{array}{l} {(2 x)}^{2} & 2 x^{2} \\ {(2 \cdot 5)}^{2} & 2 \cdot 5^{2} \\ {(10)}^{2} & 2 \cdot 25 \\ 100 & \neq & 50 \end{array}

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

Practice set c

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

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

Reading exponential notation

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 $= x x x = x^{3}$

A cube with length of side equal to x.

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

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

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

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

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

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

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

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

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

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

Practice set d

Use the order of operations to simplify the following.

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

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

$1^{4} + {(2^{2} + 4)}^{2} \div 2^{3}$

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

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

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

$x^{8}$

$(- 3)$ cubed

5 times $s$ squared

$5 s^{2}$

3 squared times $y$ to the fifth

$a$ cubed minus $(b + 7)$ squared

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

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

$x x x x x$

$x^{5}$

$(8) (8) x x x x$

$2 \cdot 3 \cdot 3 \cdot 3 \cdot 3 x x y y y y y$

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

$2 \cdot 2 \cdot 5 \cdot 6 \cdot 6 \cdot 6 x y y z z z w w w w$

$7 x x (a + 8) (a + 8)$

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

$10 x y y (c + 5) (c + 5) (c + 5)$

$4 x 4 x 4 x 4 x 4 x$

${(4 x)}^{5} or 4^{5} x^{5}$

$(9 a) (9 a) (9 a) (9 a)$

$(- 7) (- 7) (- 7) a a b b b a (- 7) b a a b$

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

$(a - 10) (a - 10) (a + 10)$

$(z + w) (z + w) (z + w) (z - w) (z - w)$

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

$(2 y) (2 y) 2 y 2 y$

$3 x y x x y - (x + 1) (x + 1) (x + 1)$

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

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

$4^{3}$

$6^{2}$

$6 \cdot 6$

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

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

$8 \cdot x \cdot x \cdot x \cdot y \cdot y$

${(18 x^{2} y^{4})}^{2}$

${(9 a^{3} b^{2})}^{3}$

$(9 a a a b b) (9 a a a b b) (9 a a a b b) or 9 \cdot 9 \cdot 9 a a a a a a a a a b b b b b b$

$5 x^{2} {(2 y^{3})}^{3}$

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

$10 a a a b b (3 c) (3 c) or 10 \cdot 3 \cdot 3 a a a b b c c$

${(a + 10)}^{2} {(a^{2} + 10)}^{2}$

$(x^{2} - y^{2}) (x^{2} + y^{2})$

$(x x - y y) (x x + y y)$

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

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

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

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

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

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

zero

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

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

$3^{2} + 7$

$4^{3} - 18$

$5^{2} + 2 (40)$

105

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

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

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

$(6^{2} - 4^{2}) \div 5$

$2^{2} (10 - 2^{3})$

$(3^{4} - 4^{3}) \div 17$

${(4 + 3)}^{2} + 1 \div (2 \cdot 5)$

${(2^{4} + 2^{5} - 2^{3} \cdot 5)}^{2} \div 4^{2}$

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

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

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

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

$\frac{51}{19}$

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

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

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

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

$\frac{1070}{11} or 97. \bar{27}$

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.

A number line with arrows on each end, labeled from negative five to five in increments of one. There is a closed circle at two, and an open circle at negative three. These circles are connected by a black line.

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

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

$x y$

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

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Source: OpenStax, Algebra i for the community college. OpenStax CNX. Dec 19, 2014 Download for free at http://legacy.cnx.org/content/col11598/1.3

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7.1 Basic properties of exponents (Page 2/2)

Practice set b

Sample set c

Practice set c

Reading exponential notation

Base

Exponent

Power

x To the n Th power

x Squared and x Cubed

The order of operations

The order of operations

Sample set d

Practice set d

Exercises

Exercises for review

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$x$ To the $n$ Th power

$x$ Squared and $x$ Cubed