Exercise 99 Page 284

a Let's name the number we need to multiply

8

by to get

1

n .

We want to know what we should multiply

n

by to get

1 .

We can write the following equation.

8 n = 1

By solving this equation for

n,

we can determine what this number is.

8 n = 1

DivEqn

$LHS / 8 = RHS / 8$

n = \frac{1}{8}

b Like in Part A, we will write an equation containing the product of

x

and our number

n

on one side and

1

on the other side.

n x = 1

By solving this equation for

n,

we can determine what this number is.

n x = 1

DivEqn

$LHS / x = RHS / x$

n = \frac{1}{x}

c Examining the equation, we see that

m

is raised to the power of

8 .

To solve for

m

we have to eliminate the exponent. We can do that by taking the

8^{th}

root of both sides. Notice that because the exponent is even we get two solutions.

8 m^{8} = \pm 840 ⇕ m = \pm 840

However many, if not all, calculators do not have the

8^{th}

root as a button. Instead, we can raise both sides to the inverse of

8,

which is the equivalent to taking the

8^{th}

root.

(m^{8})^{\frac{1}{8}} = \pm (40)^{\frac{1}{8}} ⇕ m = \pm (40)^{\frac{1}{8}}

Using a calculator, we can determine the right-hand side.

As we can see, $m \approx \pm 1.5858 \dots .$ We can check it by raising the two solutions to the power of $8$ on our calculator.

d From Part C, we know that to eliminate an exponent we have to raise the power to the inverse of the exponent. Additionally, if the exponent is even, we get a positive and a negative solution.

(n^{6})^{\frac{1}{6}} = \pm (300)^{\frac{1}{6}} ⇕ m = \pm (300)^{\frac{1}{6}}

Again, we will use the calculator to solve this.

We get the solutions $n \approx \pm 2.59 .$

e As we talked about in previous parts, we have to raise a power to the inverse of its exponent to solve for

x .

Additionally, if

a

is even, we get a positive and a negative solution.

a is even : a is odd : x^{a} = b x = \pm b^{\frac{1}{a}} x^{a} = b x = b^{\frac{1}{a}}

Subchapter links

6.2.1 Using Logarithms to Solve Exponential Equations p.284-285

100

101

102

103

6.2.2 Investigating the Properties of Logarithms p.290-291

113

114

115

116

117

118

119

120

121

122

6.2.3 Writing Equations of Exponential Functions p.294-295

127

128

129

130

131

132

133

134

135

136

6.2.4 An Application of Logarithms p.299-301

138

139

140

141

142

143

144

145

146

147