Algebra Examples

$\sqrt{16} \div 4^{3} = 4^{n}$

Step 1

Rewrite the equation as $4^{n} = \sqrt{16} \div 4^{3}$ .

$4^{n} = \sqrt{16} \div 4^{3}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{16}$ as $16^{\frac{1}{2}}$ .

$4^{n} = 16^{\frac{1}{2}} \div 4^{3}$

Step 3

Move $4^{3}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$4^{n} = 16^{\frac{1}{2}} \cdot 4^{- 3}$

Step 4

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

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

Step 5

Multiply the exponents in ${(2^{4})}^{\frac{1}{2}}$ .

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Step 5.1

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

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

Step 5.2

Cancel the common factor of $2$ .

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Step 5.2.1

Factor $2$ out of $4$ .

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

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

Step 6

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Multiply $2$ by $- 3$ .

$4^{n} = 2^{2} \cdot 2^{- 6}$

Step 8

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

$4^{n} = 2^{2 - 6}$

Step 9

Subtract $6$ from $2$ .

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

Step 10

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

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

Step 11

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

$2 n = - 4$

Step 12

Divide each term in $2 n = - 4$ by $2$ and simplify.

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Step 12.1

Divide each term in $2 n = - 4$ by $2$ .

$\frac{2 n}{2} = \frac{- 4}{2}$

Step 12.2

Simplify the left side.

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Step 12.2.1

Cancel the common factor of $2$ .

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Step 12.2.1.1

Cancel the common factor.

$\frac{2 n}{2} = \frac{- 4}{2}$

Step 12.2.1.2

Divide $n$ by $1$ .

$n = \frac{- 4}{2}$

Step 12.3

Simplify the right side.

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Step 12.3.1

Divide $- 4$ by $2$ .

$n = - 2$