Precalculus Examples

$4^{2 n + 2} \cdot 64 = 64$

Step 1

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

${(2^{2})}^{2 n + 2} \cdot 64 = 64$

Step 2

Multiply the exponents in ${(2^{2})}^{2 n + 2}$ .

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Step 2.1

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

$2^{2 (2 n + 2)} \cdot 64 = 64$

Step 2.2

Apply the distributive property.

$2^{2 (2 n) + 2 \cdot 2} \cdot 64 = 64$

Step 2.3

Multiply $2$ by $2$ .

$2^{4 n + 2 \cdot 2} \cdot 64 = 64$

Step 2.4

Multiply $2$ by $2$ .

$2^{4 n + 4} \cdot 64 = 64$

Step 3

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

$2^{4 n + 4} \cdot 2^{6} = 64$

Step 4

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

$2^{4 n + 4 + 6} = 64$

Step 5

Add $4$ and $6$ .

$2^{4 n + 10} = 64$

Step 6

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

$2^{4 n + 10} = 2^{6}$

Step 7

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

$4 n + 10 = 6$

Step 8

Solve for $n$ .

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Step 8.1

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

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Step 8.1.1

Subtract $10$ from both sides of the equation.

$4 n = 6 - 10$

Step 8.1.2

Subtract $10$ from $6$ .

$4 n = - 4$

Step 8.2

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

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Step 8.2.1

Divide each term in $4 n = - 4$ by $4$ .

$\frac{4 n}{4} = \frac{- 4}{4}$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Cancel the common factor of $4$ .

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Step 8.2.2.1.1

Cancel the common factor.

$\frac{4 n}{4} = \frac{- 4}{4}$

Step 8.2.2.1.2

Divide $n$ by $1$ .

$n = \frac{- 4}{4}$

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Divide $- 4$ by $4$ .

$n = - 1$