Basic Math Examples

${(\frac{1}{8})}^{3 - n} \cdot 4 = 4$

Step 1

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

$\frac{1^{3 - n}}{8^{3 - n}} \cdot 4 = 4$

Step 2

One to any power is one.

$\frac{1}{8^{3 - n}} \cdot 4 = 4$

Step 3

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

$8^{- (3 - n)} \cdot 4 = 4$

Step 4

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

${(2^{3})}^{- (3 - n)} \cdot 4 = 4$

Step 5

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

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Step 5.1

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

$2^{3 (- (3 - n))} \cdot 4 = 4$

Step 5.2

Apply the distributive property.

$2^{3 (- 1 \cdot 3 - - n)} \cdot 4 = 4$

Step 5.3

Multiply $- 1$ by $3$ .

$2^{3 (- 3 - - n)} \cdot 4 = 4$

Step 5.4

Multiply $- - n$ .

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Step 5.4.1

Multiply $- 1$ by $- 1$ .

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

Step 5.4.2

Multiply $n$ by $1$ .

$2^{3 (- 3 + n)} \cdot 4 = 4$

Step 5.5

Apply the distributive property.

$2^{3 \cdot - 3 + 3 n} \cdot 4 = 4$

Step 5.6

Multiply $3$ by $- 3$ .

$2^{- 9 + 3 n} \cdot 4 = 4$

Step 6

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

$2^{- 9 + 3 n} \cdot 2^{2} = 4$

Step 7

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

$2^{- 9 + 3 n + 2} = 4$

Step 8

Add $- 9$ and $2$ .

$2^{3 n - 7} = 4$

Step 9

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

$2^{3 n - 7} = 2^{2}$

Step 10

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

$3 n - 7 = 2$

Step 11

Solve for $n$ .

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Step 11.1

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

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Step 11.1.1

Add $7$ to both sides of the equation.

$3 n = 2 + 7$

Step 11.1.2

Add $2$ and $7$ .

$3 n = 9$

Step 11.2

Divide each term in $3 n = 9$ by $3$ and simplify.

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Step 11.2.1

Divide each term in $3 n = 9$ by $3$ .

$\frac{3 n}{3} = \frac{9}{3}$

Step 11.2.2

Simplify the left side.

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Step 11.2.2.1

Cancel the common factor of $3$ .

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Step 11.2.2.1.1

Cancel the common factor.

$\frac{3 n}{3} = \frac{9}{3}$

Step 11.2.2.1.2

Divide $n$ by $1$ .

$n = \frac{9}{3}$

Step 11.2.3

Simplify the right side.

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Step 11.2.3.1

Divide $9$ by $3$ .

$n = 3$