Finite Math Examples

$p {(\sqrt{\frac{2}{3} \cdot (40)})}^{2} (\frac{2 (40)}{\sqrt{3}})$

Step 1

Multiply $2$ by $40$ .

$p {(\sqrt{\frac{2}{3} \cdot (40)})}^{2} \frac{80}{\sqrt{3}}$

Step 2

Combine $\frac{2}{3}$ and $40$ .

$p {\sqrt{\frac{2 \cdot 40}{3}}}^{2} \frac{80}{\sqrt{3}}$

Step 3

Multiply $2$ by $40$ .

$p {\sqrt{\frac{80}{3}}}^{2} \frac{80}{\sqrt{3}}$

Step 4

Rewrite $\sqrt{\frac{80}{3}}$ as $\frac{\sqrt{80}}{\sqrt{3}}$ .

$p {(\frac{\sqrt{80}}{\sqrt{3}})}^{2} \frac{80}{\sqrt{3}}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $80$ as $4^{2} \cdot 5$ .

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Step 5.1.1

Factor $16$ out of $80$ .

$p {(\frac{\sqrt{16 (5)}}{\sqrt{3}})}^{2} \frac{80}{\sqrt{3}}$

Step 5.1.2

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

$p {(\frac{\sqrt{4^{2} \cdot 5}}{\sqrt{3}})}^{2} \frac{80}{\sqrt{3}}$

Step 5.2

Pull terms out from under the radical.

$p {(\frac{4 \sqrt{5}}{\sqrt{3}})}^{2} \frac{80}{\sqrt{3}}$

Step 6

Multiply $\frac{4 \sqrt{5}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$p {(\frac{4 \sqrt{5}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})}^{2} \frac{80}{\sqrt{3}}$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{4 \sqrt{5}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$p {(\frac{4 \sqrt{5} \sqrt{3}}{\sqrt{3} \sqrt{3}})}^{2} \frac{80}{\sqrt{3}}$

Step 7.2

Raise $\sqrt{3}$ to the power of $1$ .

$p {(\frac{4 \sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}})}^{2} \frac{80}{\sqrt{3}}$

Step 7.3

Raise $\sqrt{3}$ to the power of $1$ .

$p {(\frac{4 \sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}})}^{2} \frac{80}{\sqrt{3}}$

Step 7.4

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

$p {(\frac{4 \sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1 + 1}})}^{2} \frac{80}{\sqrt{3}}$

Step 7.5

Add $1$ and $1$ .

$p {(\frac{4 \sqrt{5} \sqrt{3}}{{\sqrt{3}}^{2}})}^{2} \frac{80}{\sqrt{3}}$

Step 7.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$p {(\frac{4 \sqrt{5} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}})}^{2} \frac{80}{\sqrt{3}}$

Step 7.6.2

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

$p {(\frac{4 \sqrt{5} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}})}^{2} \frac{80}{\sqrt{3}}$

Step 7.6.3

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

$p {(\frac{4 \sqrt{5} \sqrt{3}}{3^{\frac{2}{2}}})}^{2} \frac{80}{\sqrt{3}}$

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

$p {(\frac{4 \sqrt{5} \sqrt{3}}{3^{\frac{2}{2}}})}^{2} \frac{80}{\sqrt{3}}$

Step 7.6.4.2

Rewrite the expression.

$p {(\frac{4 \sqrt{5} \sqrt{3}}{3^{1}})}^{2} \frac{80}{\sqrt{3}}$

Step 7.6.5

Evaluate the exponent.

$p {(\frac{4 \sqrt{5} \sqrt{3}}{3})}^{2} \frac{80}{\sqrt{3}}$

Step 8

Simplify the numerator.

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Step 8.1

Combine using the product rule for radicals.

$p {(\frac{4 \sqrt{3 \cdot 5}}{3})}^{2} \frac{80}{\sqrt{3}}$

Step 8.2

Multiply $3$ by $5$ .

$p {(\frac{4 \sqrt{15}}{3})}^{2} \frac{80}{\sqrt{3}}$

Step 9

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 9.1

Apply the product rule to $\frac{4 \sqrt{15}}{3}$ .

$p \frac{{(4 \sqrt{15})}^{2}}{3^{2}} \frac{80}{\sqrt{3}}$

Step 9.2

Apply the product rule to $4 \sqrt{15}$ .

$p \frac{4^{2} {\sqrt{15}}^{2}}{3^{2}} \frac{80}{\sqrt{3}}$

Step 10

Simplify the numerator.

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Step 10.1

Raise $4$ to the power of $2$ .

$p \frac{16 {\sqrt{15}}^{2}}{3^{2}} \frac{80}{\sqrt{3}}$

Step 10.2

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 10.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$p \frac{16 {(15^{\frac{1}{2}})}^{2}}{3^{2}} \frac{80}{\sqrt{3}}$

Step 10.2.2

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

$p \frac{16 \cdot 15^{\frac{1}{2} \cdot 2}}{3^{2}} \frac{80}{\sqrt{3}}$

Step 10.2.3

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

$p \frac{16 \cdot 15^{\frac{2}{2}}}{3^{2}} \frac{80}{\sqrt{3}}$

Step 10.2.4

Cancel the common factor of $2$ .

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Step 10.2.4.1

Cancel the common factor.

$p \frac{16 \cdot 15^{\frac{2}{2}}}{3^{2}} \frac{80}{\sqrt{3}}$

Step 10.2.4.2

Rewrite the expression.

$p \frac{16 \cdot 15^{1}}{3^{2}} \frac{80}{\sqrt{3}}$

Step 10.2.5

Evaluate the exponent.

$p \frac{16 \cdot 15}{3^{2}} \frac{80}{\sqrt{3}}$

Step 11

Simplify terms.

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Step 11.1

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

$p \frac{16 \cdot 15}{9} \frac{80}{\sqrt{3}}$

Step 11.2

Multiply $16$ by $15$ .

$p \frac{240}{9} \frac{80}{\sqrt{3}}$

Step 11.3

Cancel the common factor of $240$ and $9$ .

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Step 11.3.1

Factor $3$ out of $240$ .

$p \frac{3 (80)}{9} \frac{80}{\sqrt{3}}$

Step 11.3.2

Cancel the common factors.

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Step 11.3.2.1

Factor $3$ out of $9$ .

$p \frac{3 \cdot 80}{3 \cdot 3} \frac{80}{\sqrt{3}}$

Step 11.3.2.2

Cancel the common factor.

$p \frac{3 \cdot 80}{3 \cdot 3} \frac{80}{\sqrt{3}}$

Step 11.3.2.3

Rewrite the expression.

$p \frac{80}{3} \frac{80}{\sqrt{3}}$

Step 11.4

Combine $p$ and $\frac{80}{3}$ .

$\frac{p \cdot 80}{3} \cdot \frac{80}{\sqrt{3}}$

Step 11.5

Move $80$ to the left of $p$ .

$\frac{80 \cdot p}{3} \cdot \frac{80}{\sqrt{3}}$

Step 12

Multiply $\frac{80}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{80 p}{3} (\frac{80}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 13

Combine and simplify the denominator.

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Step 13.1

Multiply $\frac{80}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{80 p}{3} \cdot \frac{80 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 13.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{80 p}{3} \cdot \frac{80 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 13.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{80 p}{3} \cdot \frac{80 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 13.4

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

$\frac{80 p}{3} \cdot \frac{80 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 13.5

Add $1$ and $1$ .

$\frac{80 p}{3} \cdot \frac{80 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 13.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 13.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{80 p}{3} \cdot \frac{80 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 13.6.2

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

$\frac{80 p}{3} \cdot \frac{80 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 13.6.3

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

$\frac{80 p}{3} \cdot \frac{80 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 13.6.4

Cancel the common factor of $2$ .

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Step 13.6.4.1

Cancel the common factor.

$\frac{80 p}{3} \cdot \frac{80 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 13.6.4.2

Rewrite the expression.

$\frac{80 p}{3} \cdot \frac{80 \sqrt{3}}{3^{1}}$

Step 13.6.5

Evaluate the exponent.

$\frac{80 p}{3} \cdot \frac{80 \sqrt{3}}{3}$

Step 14

Multiply $\frac{80 p}{3} \cdot \frac{80 \sqrt{3}}{3}$ .

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Step 14.1

Multiply $\frac{80 p}{3}$ by $\frac{80 \sqrt{3}}{3}$ .

$\frac{80 p (80 \sqrt{3})}{3 \cdot 3}$

Step 14.2

Multiply $80$ by $80$ .

$\frac{6400 p \sqrt{3}}{3 \cdot 3}$

Step 14.3

Multiply $3$ by $3$ .

$\frac{6400 p \sqrt{3}}{9}$