Finite Math Examples

$\sqrt{\frac{d^{3}}{7}} \cdot \sqrt{\frac{d^{7}}{7}}$

Step 1

Rewrite $\sqrt{\frac{d^{3}}{7}}$ as $\frac{\sqrt{d^{3}}}{\sqrt{7}}$ .

$\frac{\sqrt{d^{3}}}{\sqrt{7}} \cdot \sqrt{\frac{d^{7}}{7}}$

Step 2

Simplify the numerator.

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Step 2.1

Factor out $d^{2}$ .

$\frac{\sqrt{d^{2} d}}{\sqrt{7}} \cdot \sqrt{\frac{d^{7}}{7}}$

Step 2.2

Pull terms out from under the radical.

$\frac{d \sqrt{d}}{\sqrt{7}} \cdot \sqrt{\frac{d^{7}}{7}}$

Step 3

Multiply $\frac{d \sqrt{d}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{d \sqrt{d}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}} \cdot \sqrt{\frac{d^{7}}{7}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{d \sqrt{d}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{d \sqrt{d} \sqrt{7}}{\sqrt{7} \sqrt{7}} \cdot \sqrt{\frac{d^{7}}{7}}$

Step 4.2

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{d \sqrt{d} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}} \cdot \sqrt{\frac{d^{7}}{7}}$

Step 4.3

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{d \sqrt{d} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}} \cdot \sqrt{\frac{d^{7}}{7}}$

Step 4.4

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

$\frac{d \sqrt{d} \sqrt{7}}{{\sqrt{7}}^{1 + 1}} \cdot \sqrt{\frac{d^{7}}{7}}$

Step 4.5

Add $1$ and $1$ .

$\frac{d \sqrt{d} \sqrt{7}}{{\sqrt{7}}^{2}} \cdot \sqrt{\frac{d^{7}}{7}}$

Step 4.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{d \sqrt{d} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}} \cdot \sqrt{\frac{d^{7}}{7}}$

Step 4.6.2

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

$\frac{d \sqrt{d} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}} \cdot \sqrt{\frac{d^{7}}{7}}$

Step 4.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{d \sqrt{d} \sqrt{7}}{7^{\frac{2}{2}}} \cdot \sqrt{\frac{d^{7}}{7}}$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$\frac{d \sqrt{d} \sqrt{7}}{7^{\frac{2}{2}}} \cdot \sqrt{\frac{d^{7}}{7}}$

Step 4.6.4.2

Rewrite the expression.

$\frac{d \sqrt{d} \sqrt{7}}{7^{1}} \cdot \sqrt{\frac{d^{7}}{7}}$

Step 4.6.5

Evaluate the exponent.

$\frac{d \sqrt{d} \sqrt{7}}{7} \cdot \sqrt{\frac{d^{7}}{7}}$

Step 5

Combine using the product rule for radicals.

$\frac{d \sqrt{7 d}}{7} \cdot \sqrt{\frac{d^{7}}{7}}$

Step 6

Rewrite $\sqrt{\frac{d^{7}}{7}}$ as $\frac{\sqrt{d^{7}}}{\sqrt{7}}$ .

$\frac{d \sqrt{7 d}}{7} \cdot \frac{\sqrt{d^{7}}}{\sqrt{7}}$

Step 7

Combine.

$\frac{d \sqrt{7 d} \sqrt{d^{7}}}{7 \sqrt{7}}$

Step 8

Combine $\sqrt{7 d}$ and $\sqrt{7}$ into a single radical.

$\frac{d \sqrt{\frac{7 d}{7}} \sqrt{d^{7}}}{7}$

Step 9

Cancel the common factor of $7$ .

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Step 9.1

Cancel the common factor.

$\frac{d \sqrt{\frac{7 d}{7}} \sqrt{d^{7}}}{7}$

Step 9.2

Divide $d$ by $1$ .

$\frac{d \sqrt{d} \sqrt{d^{7}}}{7}$

Step 10

Simplify the numerator.

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Step 10.1

Combine using the product rule for radicals.

$\frac{d \sqrt{d^{7} d}}{7}$

Step 10.2

Multiply $d^{7}$ by $d$ by adding the exponents.

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Step 10.2.1

Multiply $d^{7}$ by $d$ .

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Step 10.2.1.1

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

$\frac{d \sqrt{d^{7} d^{1}}}{7}$

Step 10.2.1.2

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

$\frac{d \sqrt{d^{7 + 1}}}{7}$

Step 10.2.2

Add $7$ and $1$ .

$\frac{d \sqrt{d^{8}}}{7}$

Step 11

Simplify the numerator.

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Step 11.1

Rewrite $d^{8}$ as ${(d^{4})}^{2}$ .

$\frac{d \sqrt{{(d^{4})}^{2}}}{7}$

Step 11.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{d \cdot d^{4}}{7}$

Step 12

Multiply $d$ by $d^{4}$ by adding the exponents.

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Step 12.1

Multiply $d$ by $d^{4}$ .

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Step 12.1.1

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

$\frac{d^{1} d^{4}}{7}$

Step 12.1.2

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

$\frac{d^{1 + 4}}{7}$

Step 12.2

Add $1$ and $4$ .

$\frac{d^{5}}{7}$