Solve for x 2^(8x-4)*16^(3x-5)=16^(9x+2)

$2^{8 x - 4} \cdot 16^{3 x - 5} = 16^{9 x + 2}$

Step 1

Rewrite $16$ as $2^{4}$ .

$2^{8 x - 4} \cdot {(2^{4})}^{3 x - 5} = 16^{9 x + 2}$

Step 2

Multiply the exponents in ${(2^{4})}^{3 x - 5}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2^{8 x - 4} \cdot 2^{4 (3 x - 5)} = 16^{9 x + 2}$

Apply the distributive property.

$2^{8 x - 4} \cdot 2^{4 (3 x) + 4 \cdot - 5} = 16^{9 x + 2}$

Multiply $3$ by $4$ .

$2^{8 x - 4} \cdot 2^{12 x + 4 \cdot - 5} = 16^{9 x + 2}$

Multiply $4$ by $- 5$ .

$2^{8 x - 4} \cdot 2^{12 x - 20} = 16^{9 x + 2}$

Step 3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2^{8 x - 4 + 12 x - 20} = 16^{9 x + 2}$

Step 4

Add $8 x$ and $12 x$ .

$2^{20 x - 4 - 20} = 16^{9 x + 2}$

Step 5

Subtract $20$ from $- 4$ .

$2^{20 x - 24} = 16^{9 x + 2}$

Step 6

Create equivalent expressions in the equation that all have equal bases.

$2^{20 x - 24} = 2^{4 (9 x + 2)}$

Step 7

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$20 x - 24 = 4 (9 x + 2)$

Step 8

Solve for $x$ .

Simplify $4 (9 x + 2)$ .

Rewrite.

$20 x - 24 = 0 + 0 + 4 (9 x + 2)$

Simplify by adding zeros.

$20 x - 24 = 4 (9 x + 2)$

Apply the distributive property.

$20 x - 24 = 4 (9 x) + 4 \cdot 2$

Multiply.

Multiply $9$ by $4$ .

$20 x - 24 = 36 x + 4 \cdot 2$

Multiply $4$ by $2$ .

$20 x - 24 = 36 x + 8$

Move all terms containing $x$ to the left side of the equation.

Subtract $36 x$ from both sides of the equation.

$20 x - 24 - 36 x = 8$

Subtract $36 x$ from $20 x$ .

$- 16 x - 24 = 8$

Move all terms not containing $x$ to the right side of the equation.

Add $24$ to both sides of the equation.

$- 16 x = 8 + 24$

Add $8$ and $24$ .

$- 16 x = 32$

Divide each term in $- 16 x = 32$ by $- 16$ and simplify.

Divide each term in $- 16 x = 32$ by $- 16$ .

$\frac{- 16 x}{- 16} = \frac{32}{- 16}$

Simplify the left side.

Cancel the common factor of $- 16$ .

Cancel the common factor.

$\frac{- 16 x}{- 16} = \frac{32}{- 16}$

Divide $x$ by $1$ .

$x = \frac{32}{- 16}$

Simplify the right side.

Divide $32$ by $- 16$ .

$x = - 2$

Solve for x 2^(8x-4)*16^(3x-5)=16^(9x+2)

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