Algebra Examples

$2^{6 x - 8} \cdot 64^{2 - x} = 2^{9 x + 4}$

Step 1

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

$2^{6 x - 8} \cdot {(2^{6})}^{2 - x} = 2^{9 x + 4}$

Step 2

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

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Step 2.1

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

$2^{6 x - 8} \cdot 2^{6 (2 - x)} = 2^{9 x + 4}$

Step 2.2

Apply the distributive property.

$2^{6 x - 8} \cdot 2^{6 \cdot 2 + 6 (- x)} = 2^{9 x + 4}$

Step 2.3

Multiply $6$ by $2$ .

$2^{6 x - 8} \cdot 2^{12 + 6 (- x)} = 2^{9 x + 4}$

Step 2.4

Multiply $- 1$ by $6$ .

$2^{6 x - 8} \cdot 2^{12 - 6 x} = 2^{9 x + 4}$

Step 3

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

$2^{6 x - 8 + 12 - 6 x} = 2^{9 x + 4}$

Step 4

Subtract $6 x$ from $6 x$ .

$2^{0 - 8 + 12} = 2^{9 x + 4}$

Step 5

Subtract $8$ from $0$ .

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

Step 6

Add $- 8$ and $12$ .

$2^{4} = 2^{9 x + 4}$

Step 7

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

$4 = 9 x + 4$

Step 8

Solve for $x$ .

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Step 8.1

Rewrite the equation as $9 x + 4 = 4$ .

$9 x + 4 = 4$

Step 8.2

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

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Step 8.2.1

Subtract $4$ from both sides of the equation.

$9 x = 4 - 4$

Step 8.2.2

Subtract $4$ from $4$ .

$9 x = 0$

Step 8.3

Divide each term in $9 x = 0$ by $9$ and simplify.

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Step 8.3.1

Divide each term in $9 x = 0$ by $9$ .

$\frac{9 x}{9} = \frac{0}{9}$

Step 8.3.2

Simplify the left side.

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Step 8.3.2.1

Cancel the common factor of $9$ .

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Step 8.3.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{0}{9}$

Step 8.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{9}$

Step 8.3.3

Simplify the right side.

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Step 8.3.3.1

Divide $0$ by $9$ .

$x = 0$