Exercise 6 Page 301

Rewrite the terms so that they have a common base.

x=1

Practice makes perfect

To solve the given exponential equation, we will start by rewriting the terms so that they have a common base. Note that 8 cannot be represented as a natural power of 4. Therefore, we will write both 4 and 8 as the powers of 2.

4^(3x) = 8^(x+1)

WritePow

Write as a power

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

PowPow

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

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

Multiply

2^(6x) = 2^(3(x+1))

Distr

Distribute 3

2^(6x) = 2^(3x+3)

Now we have two equivalent expressions with the same base. If both sides of the equation are equal, the exponents must also be equal. \

begin{gathered} 2^{6x} = 2^{3x+3} \quad \Leftrightarrow \quad 6x = 3x + 3 \end{gathered} Finally, we will solve the equation 6x = 3x + 3.

6x = 3x + 3

SubEqn

LHS-3x=RHS-3x

3x = 3

DivEqn

.LHS /3.=.RHS /3.

x=3/3

QuotOne

a/a=1

x = 1

To check our answer, we will substitute 1 for x in the given equation.

4^(3x) = 8^(x+1)

Substitute

x= 1

4^(3( 1)) ? = 8^(1+1)

MultByOne

a * 1=a

4^3 ? = 8^(1+1)

AddTerms

Add terms

4^3 ? = 8^2

CalcPow

Calculate power

64 = 64 ✓

Since substituting 1 for x in the given equation produces a true statement, x=1 is the solution to our equation.

Subchapter links

Communicate Your Answer p.299

Monitoring Progress p.300-302

Exercises p.303-304

Exercise 6 Page 301

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