Basic Math Examples

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

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

Step 1

Apply the product rule to

\frac{1}{8}

$\frac{1}{8}$ .

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

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

Step 2

One to any power is one.

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

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

Step 3

Move

8^{3 - n}

$8^{3 - n}$ to the numerator using the negative exponent rule

\frac{1}{b^{- n}} = b^{n}

$\frac{1}{b^{- n}} = b^{n}$ .

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

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

Step 4

Rewrite

8

$8$ as

2^{3}

$2^{3}$ .

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

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

Step 5

Multiply the exponents in

{(2^{3})}^{- (3 - n)}

${(2^{3})}^{- (3 - n)}$ .

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Step 5.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 5.2

Apply the distributive property.

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

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

Step 5.3

Multiply

- 1

$- 1$ by

3

$3$ .

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

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

Step 5.4

Multiply

- - n

$- - n$ .

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Step 5.4.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

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

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

Step 5.4.2

Multiply

n

$n$ by

1

$1$ .

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

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

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

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

Step 5.5

Apply the distributive property.

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

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

Step 5.6

Multiply

3

$3$ by

- 3

$- 3$ .

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

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

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

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

Step 6

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

Step 7

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 8

Add

- 9

$- 9$ and

2

$2$ .

2^{3 n - 7} = 4

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

Step 9

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

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

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

Step 10

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

3 n - 7 = 2

$3 n - 7 = 2$

Step 11

Solve for

n

$n$ .

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Step 11.1

Move all terms not containing

n

$n$ to the right side of the equation.

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Step 11.1.1

Add

7

$7$ to both sides of the equation.

3 n = 2 + 7

$3 n = 2 + 7$

Step 11.1.2

Add

2

$2$ and

7

$7$ .

3 n = 9

$3 n = 9$

3 n = 9

$3 n = 9$

Step 11.2

Divide each term in

3 n = 9

$3 n = 9$ by

3

$3$ and simplify.

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Step 11.2.1

Divide each term in

3 n = 9

$3 n = 9$ by

3

$3$ .

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

$3$ .

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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

$3$ .

$n = 3$