Solve for x 64^(3x+3)*32=2^6

$64^{3 x + 3} \cdot 32 = 2^{6}$

Step 1

Rewrite $64$ as $2^{6}$ .

${(2^{6})}^{3 x + 3} \cdot 32 = 2^{6}$

Step 2

Multiply the exponents in ${(2^{6})}^{3 x + 3}$ .

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

$2^{6 (3 x + 3)} \cdot 32 = 2^{6}$

Apply the distributive property.

$2^{6 (3 x) + 6 \cdot 3} \cdot 32 = 2^{6}$

Multiply $3$ by $6$ .

$2^{18 x + 6 \cdot 3} \cdot 32 = 2^{6}$

Multiply $6$ by $3$ .

$2^{18 x + 18} \cdot 32 = 2^{6}$

Step 3

Rewrite $32$ as $2^{5}$ .

$2^{18 x + 18} \cdot 2^{5} = 2^{6}$

Step 4

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

$2^{18 x + 18 + 5} = 2^{6}$

Step 5

Add $18$ and $5$ .

$2^{18 x + 23} = 2^{6}$

Step 6

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

$18 x + 23 = 6$

Step 7

Solve for $x$ .

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

Subtract $23$ from both sides of the equation.

$18 x = 6 - 23$

Subtract $23$ from $6$ .

$18 x = - 17$

Divide each term in $18 x = - 17$ by $18$ and simplify.

Divide each term in $18 x = - 17$ by $18$ .

$\frac{18 x}{18} = \frac{- 17}{18}$

Simplify the left side.

Cancel the common factor of $18$ .

Cancel the common factor.

$\frac{18 x}{18} = \frac{- 17}{18}$

Divide $x$ by $1$ .

$x = \frac{- 17}{18}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{17}{18}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{17}{18}$

Decimal Form:

$x = - 0.9\overline{4}$

Solve for x 64^(3x+3)*32=2^6

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