Precalculus Examples

$64 \cdot 16^{- 3 x} = 16^{3 x - 2}$

Step 1

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

$2^{6} \cdot 16^{- 3 x} = 16^{3 x - 2}$

Step 2

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

$2^{6} \cdot {(2^{4})}^{- 3 x} = 16^{3 x - 2}$

Step 3

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

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Step 3.1

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

$2^{6} \cdot 2^{4 (- 3 x)} = 16^{3 x - 2}$

Step 3.2

Multiply $- 3$ by $4$ .

$2^{6} \cdot 2^{- 12 x} = 16^{3 x - 2}$

Step 4

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

$2^{6 - 12 x} = 16^{3 x - 2}$

Step 5

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

$2^{6 - 12 x} = 2^{4 (3 x - 2)}$

Step 6

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

$6 - 12 x = 4 (3 x - 2)$

Step 7

Solve for $x$ .

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Step 7.1

Simplify $4 (3 x - 2)$ .

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Step 7.1.1

Rewrite.

$6 - 12 x = 0 + 0 + 4 (3 x - 2)$

Step 7.1.2

Simplify by adding zeros.

$6 - 12 x = 4 (3 x - 2)$

Step 7.1.3

Apply the distributive property.

$6 - 12 x = 4 (3 x) + 4 \cdot - 2$

Step 7.1.4

Multiply.

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Step 7.1.4.1

Multiply $3$ by $4$ .

$6 - 12 x = 12 x + 4 \cdot - 2$

Step 7.1.4.2

Multiply $4$ by $- 2$ .

$6 - 12 x = 12 x - 8$

Step 7.2

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

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Step 7.2.1

Subtract $12 x$ from both sides of the equation.

$6 - 12 x - 12 x = - 8$

Step 7.2.2

Subtract $12 x$ from $- 12 x$ .

$6 - 24 x = - 8$

Step 7.3

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

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Step 7.3.1

Subtract $6$ from both sides of the equation.

$- 24 x = - 8 - 6$

Step 7.3.2

Subtract $6$ from $- 8$ .

$- 24 x = - 14$

Step 7.4

Divide each term in $- 24 x = - 14$ by $- 24$ and simplify.

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Step 7.4.1

Divide each term in $- 24 x = - 14$ by $- 24$ .

$\frac{- 24 x}{- 24} = \frac{- 14}{- 24}$

Step 7.4.2

Simplify the left side.

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Step 7.4.2.1

Cancel the common factor of $- 24$ .

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Step 7.4.2.1.1

Cancel the common factor.

$\frac{- 24 x}{- 24} = \frac{- 14}{- 24}$

Step 7.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 14}{- 24}$

Step 7.4.3

Simplify the right side.

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Step 7.4.3.1

Cancel the common factor of $- 14$ and $- 24$ .

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Step 7.4.3.1.1

Factor $- 2$ out of $- 14$ .

$x = \frac{- 2 (7)}{- 24}$

Step 7.4.3.1.2

Cancel the common factors.

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Step 7.4.3.1.2.1

Factor $- 2$ out of $- 24$ .

$x = \frac{- 2 \cdot 7}{- 2 \cdot 12}$

Step 7.4.3.1.2.2

Cancel the common factor.

$x = \frac{- 2 \cdot 7}{- 2 \cdot 12}$

Step 7.4.3.1.2.3

Rewrite the expression.

$x = \frac{7}{12}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7}{12}$

Decimal Form:

$x = 0.58\overline{3}$