Solve for x (1/8)^(3x)*64^(2x+1)=4

${(\frac{1}{8})}^{3 x} \cdot 64^{2 x + 1} = 4$

Step 1

Apply the product rule to $\frac{1}{8}$ .

$\frac{1^{3 x}}{8^{3 x}} \cdot 64^{2 x + 1} = 4$

Step 2

One to any power is one.

$\frac{1}{8^{3 x}} \cdot 64^{2 x + 1} = 4$

Step 3

Move $8^{3 x}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$8^{- 3 x} \cdot 64^{2 x + 1} = 4$

Step 4

Rewrite $8$ as $2^{3}$ .

${(2^{3})}^{- 3 x} \cdot 64^{2 x + 1} = 4$

Step 5

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

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

$2^{3 (- 3 x)} \cdot 64^{2 x + 1} = 4$

Multiply $- 3$ by $3$ .

$2^{- 9 x} \cdot 64^{2 x + 1} = 4$

Step 6

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

$2^{- 9 x} \cdot {(2^{6})}^{2 x + 1} = 4$

Step 7

Multiply the exponents in ${(2^{6})}^{2 x + 1}$ .

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

$2^{- 9 x} \cdot 2^{6 (2 x + 1)} = 4$

Apply the distributive property.

$2^{- 9 x} \cdot 2^{6 (2 x) + 6 \cdot 1} = 4$

Multiply $2$ by $6$ .

$2^{- 9 x} \cdot 2^{12 x + 6 \cdot 1} = 4$

Multiply $6$ by $1$ .

$2^{- 9 x} \cdot 2^{12 x + 6} = 4$

Step 8

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

$2^{- 9 x + 12 x + 6} = 4$

Step 9

Add $- 9 x$ and $12 x$ .

$2^{3 x + 6} = 4$

Step 10

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

$2^{3 x + 6} = 2^{2}$

Step 11

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

$3 x + 6 = 2$

Step 12

Solve for $x$ .

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

Subtract $6$ from both sides of the equation.

$3 x = 2 - 6$

Subtract $6$ from $2$ .

$3 x = - 4$

Divide each term in $3 x = - 4$ by $3$ and simplify.

Divide each term in $3 x = - 4$ by $3$ .

$\frac{3 x}{3} = \frac{- 4}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 4}{3}$

Divide $x$ by $1$ .

$x = \frac{- 4}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{4}{3}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{4}{3}$

Decimal Form:

$x = - 1.\overline{3}$

Mixed Number Form:

$x = - 1 \frac{1}{3}$

Solve for x (1/8)^(3x)*64^(2x+1)=4

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