Exercise 12 Page 303

Rewrite the terms so that they have a common base.

x=-4

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 27 cannot be represented as a natural power of 9. Therefore, we will write both 27 and 9 as powers of 3.

27^x = 9^(x-2)

WritePow

Write as a power

( 3^3 )^x = ( 3^2 )^(x-2)

PowPow

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

3^(3x) = 3^(2(x-2))

Distr

Distribute 2

3^(3x) = 3^(2x-4)

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^(3x) = 3^(2x-4) ⇔ 3x = 2x-4 Finally, we will solve the equation 3x = 2x-4 by subtracting 2x from both sides of the equation. 3x = 2x-4 ⇔ x = -4 To check our answer, we will substitute -4 for x in the given equation.

27^x = 9^(x-2)

Substitute

x= -4

27^(-4) ? = 9^(-4-2)

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Simplify

SubTerm

Subtract term

27^(-4) ? = 9^(-6)

WritePow

Write as a power

( 3^3 )^(-4) ? = ( 3^2 )^(-6)