Exercise 53 Page 488

Practice makes perfect

a To solve the given exponential equation, we will start by rewriting the terms so that they have the same base.

8^x=2^6

WritePow

Write as a power

( 2^3 )^x=2^6

PowPow

(a^m)^n=a^(m* n)

2^(3x)=2^6

Now, we have two equivalent expressions with the same base. If both sides of the equation are equal, the exponents must also be equal. 2^(3x)=2^6 ⇔ 3x=6 Finally, we will solve the equation 3x=6.

3x=6

DivEqn

.LHS /3.=.RHS /3.

x=2

b To solve the given exponential equation, we will start by rewriting the terms so that they have the same base.

9^2=3^(2x+1)

WritePow

Write as a power

( 3^2 )^2=3^(2x+1)

PowPow

(a^m)^n=a^(m* n)

3^4=3^(2x+1)

Now, we have two equivalent expressions with the same base. If both sides of the equation are equal, the exponents must also be equal. 3^4=3^(2x+1) ⇔ 4=2x+1 Finally, we will solve the equation 4=2x+1.

4=2x+1

▼

Solve for x

SubEqn

LHS-1=RHS-1

3=2x

RearrangeEqn

Rearrange equation

2x=3

DivEqn

.LHS /2.=.RHS /2.

x=1.5

c To solve the given exponential equation, we will start by rewriting the terms so that they have the same base. Recall that 1a=a^(- 1).

4^(2x)=( 1/2 )^(x+5)

WritePow

Write as a power

( 2^2 )^(2x)=( 1/2 )^(x+5)

1/a=a^(- 1)

( 2^2 )^(2x)=( 2^(- 1) )^(x+5)

PowPow

(a^m)^n=a^(m* n)

2^(4x)=2^(- x -5)

Now, we have two equivalent expressions with the same base. If both sides of the equation are equal, the exponents must also be equal. 2^(4x)=2^(- x -5) ⇔ 4x=- x-5 Finally, we will solve the equation 4x=- x-5.

4x=- x-5

▼

Solve for x

AddEqn

LHS+x=RHS+x

5x=- 5

DivEqn

.LHS /5.=.RHS /5.

x=- 1

Subchapter links

10.2.1 Solving by Rewriting p.479-480

10.2.2 Fraction Busters p.483-484

10.2.3 Multiple Methods for Solving Equations p.488-489

10.2.4 Determining the Number of Solutions p.493-494

10.2.5 Deriving the Quadratic Formula and the Number System p.499-500

10.2.6 More Solving and an Application p.502-503

Exercise 53 Page 488

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