Precalculus Examples

$r = \sqrt{\frac{7 S}{p \cdot p}}$

Step 1

Rewrite the equation as $\sqrt{\frac{7 S}{p \cdot p}} = r$ .

$\sqrt{\frac{7 S}{p \cdot p}} = r$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{\frac{7 S}{p \cdot p}}}^{2} = r^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

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

${({(\frac{7 S}{p \cdot p})}^{\frac{1}{2}})}^{2} = r^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(\frac{7 S}{p \cdot p})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(\frac{7 S}{p \cdot p})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(\frac{7 S}{p \cdot p})}^{\frac{1}{2} \cdot 2} = r^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(\frac{7 S}{p \cdot p})}^{\frac{1}{2} \cdot 2} = r^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(\frac{7 S}{p \cdot p})}^{1} = r^{2}$

Step 3.2.1.2

Multiply $p$ by $p$ .

${(\frac{7 S}{p^{2}})}^{1} = r^{2}$

Step 3.2.1.3

Simplify.

$\frac{7 S}{p^{2}} = r^{2}$

Step 4

Solve for $S$ .

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Step 4.1

Multiply both sides by $p^{2}$ .

$\frac{7 S}{p^{2}} p^{2} = r^{2} p^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $p^{2}$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{7 S}{p^{2}} p^{2} = r^{2} p^{2}$

Step 4.2.1.2

Rewrite the expression.

$7 S = r^{2} p^{2}$

Step 4.3

Divide each term in $7 S = r^{2} p^{2}$ by $7$ and simplify.

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Step 4.3.1

Divide each term in $7 S = r^{2} p^{2}$ by $7$ .

$\frac{7 S}{7} = \frac{r^{2} p^{2}}{7}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $7$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{7 S}{7} = \frac{r^{2} p^{2}}{7}$

Step 4.3.2.1.2

Divide $S$ by $1$ .

$S = \frac{r^{2} p^{2}}{7}$