Solve for x 64^(x-6)*4^(9x-3)=64^(6x+4)

$64^{x - 6} \cdot 4^{9 x - 3} = 64^{6 x + 4}$

Step 1

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

${(2^{6})}^{x - 6} \cdot 4^{9 x - 3} = 64^{6 x + 4}$

Step 2

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

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

$2^{6 (x - 6)} \cdot 4^{9 x - 3} = 64^{6 x + 4}$

Apply the distributive property.

$2^{6 x + 6 \cdot - 6} \cdot 4^{9 x - 3} = 64^{6 x + 4}$

Multiply $6$ by $- 6$ .

$2^{6 x - 36} \cdot 4^{9 x - 3} = 64^{6 x + 4}$

Step 3

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

$2^{6 x - 36} \cdot {(2^{2})}^{9 x - 3} = 64^{6 x + 4}$

Step 4

Multiply the exponents in ${(2^{2})}^{9 x - 3}$ .

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

$2^{6 x - 36} \cdot 2^{2 (9 x - 3)} = 64^{6 x + 4}$

Apply the distributive property.

$2^{6 x - 36} \cdot 2^{2 (9 x) + 2 \cdot - 3} = 64^{6 x + 4}$

Multiply $9$ by $2$ .

$2^{6 x - 36} \cdot 2^{18 x + 2 \cdot - 3} = 64^{6 x + 4}$

Multiply $2$ by $- 3$ .

$2^{6 x - 36} \cdot 2^{18 x - 6} = 64^{6 x + 4}$

Step 5

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

$2^{6 x - 36 + 18 x - 6} = 64^{6 x + 4}$

Step 6

Add $6 x$ and $18 x$ .

$2^{24 x - 36 - 6} = 64^{6 x + 4}$

Step 7

Subtract $6$ from $- 36$ .

$2^{24 x - 42} = 64^{6 x + 4}$

Step 8

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

$2^{24 x - 42} = 2^{6 (6 x + 4)}$

Step 9

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

$24 x - 42 = 6 (6 x + 4)$

Step 10

Solve for $x$ .

Simplify $6 (6 x + 4)$ .

Rewrite.

$24 x - 42 = 0 + 0 + 6 (6 x + 4)$

Simplify by adding zeros.

$24 x - 42 = 6 (6 x + 4)$

Apply the distributive property.

$24 x - 42 = 6 (6 x) + 6 \cdot 4$

Multiply.

Multiply $6$ by $6$ .

$24 x - 42 = 36 x + 6 \cdot 4$

Multiply $6$ by $4$ .

$24 x - 42 = 36 x + 24$

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

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

$24 x - 42 - 36 x = 24$

Subtract $36 x$ from $24 x$ .

$- 12 x - 42 = 24$

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

Add $42$ to both sides of the equation.

$- 12 x = 24 + 42$

Add $24$ and $42$ .

$- 12 x = 66$

Divide each term in $- 12 x = 66$ by $- 12$ and simplify.

Divide each term in $- 12 x = 66$ by $- 12$ .

$\frac{- 12 x}{- 12} = \frac{66}{- 12}$

Simplify the left side.

Cancel the common factor of $- 12$ .

Cancel the common factor.

$\frac{- 12 x}{- 12} = \frac{66}{- 12}$

Divide $x$ by $1$ .

$x = \frac{66}{- 12}$

Simplify the right side.

Cancel the common factor of $66$ and $- 12$ .

Factor $6$ out of $66$ .

$x = \frac{6 (11)}{- 12}$

Cancel the common factors.

Factor $6$ out of $- 12$ .

$x = \frac{6 \cdot 11}{6 \cdot - 2}$

Cancel the common factor.

$x = \frac{6 \cdot 11}{6 \cdot - 2}$

Rewrite the expression.

$x = \frac{11}{- 2}$

Move the negative in front of the fraction.

$x = - \frac{11}{2}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{11}{2}$

Decimal Form:

$x = - 5.5$

Mixed Number Form:

$x = - 5 \frac{1}{2}$

Solve for x 64^(x-6)*4^(9x-3)=64^(6x+4)

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