Finite Math Examples

$4 \sqrt{4 x} - \sqrt[8]{x^{4}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x}$ as ${(4 x)}^{\frac{1}{2}}$ .

$4 {(4 x)}^{\frac{1}{2}} - \sqrt[8]{x^{4}}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[8]{x^{4}}$ as $x^{\frac{4}{8}}$ .

$4 {(4 x)}^{\frac{1}{2}} - x^{\frac{4}{8}}$

Step 3

Cancel the common factor of $4$ and $8$ .

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Step 3.1

Factor $4$ out of $4$ .

$4 {(4 x)}^{\frac{1}{2}} - x^{\frac{4 (1)}{8}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $4$ out of $8$ .

$4 {(4 x)}^{\frac{1}{2}} - x^{\frac{4 \cdot 1}{4 \cdot 2}}$

Step 3.2.2

Cancel the common factor.

$4 {(4 x)}^{\frac{1}{2}} - x^{\frac{4 \cdot 1}{4 \cdot 2}}$

Step 3.2.3

Rewrite the expression.

$4 {(4 x)}^{\frac{1}{2}} - x^{\frac{1}{2}}$

Step 4

Apply the product rule to $4 x$ .

$4 (4^{\frac{1}{2}} x^{\frac{1}{2}}) - x^{\frac{1}{2}}$

Step 5

Multiply $4$ by $4^{\frac{1}{2}}$ by adding the exponents.

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Step 5.1

Move $4^{\frac{1}{2}}$ .

$4^{\frac{1}{2}} \cdot 4 x^{\frac{1}{2}} - x^{\frac{1}{2}}$

Step 5.2

Multiply $4^{\frac{1}{2}}$ by $4$ .

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Step 5.2.1

Raise $4$ to the power of $1$ .

$4^{\frac{1}{2}} \cdot 4^{1} x^{\frac{1}{2}} - x^{\frac{1}{2}}$

Step 5.2.2

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

$4^{\frac{1}{2} + 1} x^{\frac{1}{2}} - x^{\frac{1}{2}}$

Step 5.3

Write $1$ as a fraction with a common denominator.

$4^{\frac{1}{2} + \frac{2}{2}} x^{\frac{1}{2}} - x^{\frac{1}{2}}$

Step 5.4

Combine the numerators over the common denominator.

$4^{\frac{1 + 2}{2}} x^{\frac{1}{2}} - x^{\frac{1}{2}}$

Step 5.5

Add $1$ and $2$ .

$4^{\frac{3}{2}} x^{\frac{1}{2}} - x^{\frac{1}{2}}$

Step 6

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

${(2^{2})}^{\frac{3}{2}} x^{\frac{1}{2}} - x^{\frac{1}{2}}$

Step 7

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

$2^{2 (\frac{3}{2})} x^{\frac{1}{2}} - x^{\frac{1}{2}}$

Step 8

Cancel the common factor of $2$ .

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Step 8.1

Cancel the common factor.

$2^{2 (\frac{3}{2})} x^{\frac{1}{2}} - x^{\frac{1}{2}}$

Step 8.2

Rewrite the expression.

$2^{3} x^{\frac{1}{2}} - x^{\frac{1}{2}}$

Step 9

Raise $2$ to the power of $3$ .

$8 x^{\frac{1}{2}} - x^{\frac{1}{2}}$

Step 10

Subtract $x^{\frac{1}{2}}$ from $8 x^{\frac{1}{2}}$ .

$7 x^{\frac{1}{2}}$