Algebra Examples

$T = π \sqrt{\frac{L}{245}}$

Step 1

Rewrite the equation as $π \sqrt{\frac{L}{245}} = T$ .

$π \sqrt{\frac{L}{245}} = T$

Step 2

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

${(π \sqrt{\frac{L}{245}})}^{2} = T^{2}$

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 $\sqrt{\frac{L}{245}}$ as ${(\frac{L}{245})}^{\frac{1}{2}}$ .

${(π {(\frac{L}{245})}^{\frac{1}{2}})}^{2} = T^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(π {(\frac{L}{245})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Apply the product rule to $\frac{L}{245}$ .

${(π \frac{L^{\frac{1}{2}}}{245^{\frac{1}{2}}})}^{2} = T^{2}$

Step 3.2.1.2

Combine $π$ and $\frac{L^{\frac{1}{2}}}{245^{\frac{1}{2}}}$ .

${(\frac{π L^{\frac{1}{2}}}{245^{\frac{1}{2}}})}^{2} = T^{2}$

Step 3.2.1.3

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

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Step 3.2.1.3.1

Apply the product rule to $\frac{π L^{\frac{1}{2}}}{245^{\frac{1}{2}}}$ .

$\frac{{(π L^{\frac{1}{2}})}^{2}}{{(245^{\frac{1}{2}})}^{2}} = T^{2}$

Step 3.2.1.3.2

Apply the product rule to $π L^{\frac{1}{2}}$ .

$\frac{π^{2} {(L^{\frac{1}{2}})}^{2}}{{(245^{\frac{1}{2}})}^{2}} = T^{2}$

Step 3.2.1.4

Simplify the numerator.

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Step 3.2.1.4.1

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

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Step 3.2.1.4.1.1

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

$\frac{π^{2} L^{\frac{1}{2} \cdot 2}}{{(245^{\frac{1}{2}})}^{2}} = T^{2}$

Step 3.2.1.4.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.4.1.2.1

Cancel the common factor.

$\frac{π^{2} L^{\frac{1}{2} \cdot 2}}{{(245^{\frac{1}{2}})}^{2}} = T^{2}$

Step 3.2.1.4.1.2.2

Rewrite the expression.

$\frac{π^{2} L^{1}}{{(245^{\frac{1}{2}})}^{2}} = T^{2}$

Step 3.2.1.4.2

Simplify.

$\frac{π^{2} L}{{(245^{\frac{1}{2}})}^{2}} = T^{2}$

Step 3.2.1.5

Simplify the denominator.

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Step 3.2.1.5.1

Multiply the exponents in ${(245^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.5.1.1

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

$\frac{π^{2} L}{245^{\frac{1}{2} \cdot 2}} = T^{2}$

Step 3.2.1.5.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.5.1.2.1

Cancel the common factor.

$\frac{π^{2} L}{245^{\frac{1}{2} \cdot 2}} = T^{2}$

Step 3.2.1.5.1.2.2

Rewrite the expression.

$\frac{π^{2} L}{245^{1}} = T^{2}$

Step 3.2.1.5.2

Evaluate the exponent.

$\frac{π^{2} L}{245} = T^{2}$

Step 4

Solve for $L$ .

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Step 4.1

Multiply both sides of the equation by $\frac{245}{π^{2}}$ .

$\frac{245}{π^{2}} \cdot \frac{π^{2} L}{245} = \frac{245}{π^{2}} T^{2}$

Step 4.2

Simplify both sides of the equation.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Simplify $\frac{245}{π^{2}} \cdot \frac{π^{2} L}{245}$ .

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Step 4.2.1.1.1

Combine.

$\frac{245 (π^{2} L)}{π^{2} \cdot 245} = \frac{245}{π^{2}} T^{2}$

Step 4.2.1.1.2

Cancel the common factor of $245$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

$\frac{245 (π^{2} L)}{π^{2} \cdot 245} = \frac{245}{π^{2}} T^{2}$

Step 4.2.1.1.2.2

Rewrite the expression.

$\frac{π^{2} L}{π^{2}} = \frac{245}{π^{2}} T^{2}$

Step 4.2.1.1.3

Cancel the common factor of $π^{2}$ .

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Step 4.2.1.1.3.1

Cancel the common factor.

$\frac{π^{2} L}{π^{2}} = \frac{245}{π^{2}} T^{2}$

Step 4.2.1.1.3.2

Divide $L$ by $1$ .

$L = \frac{245}{π^{2}} T^{2}$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Combine $\frac{245}{π^{2}}$ and $T^{2}$ .

$L = \frac{245 T^{2}}{π^{2}}$