Basic Math Examples

$v = k \sqrt{\frac{F l}{m}}$

Step 1

Rewrite the equation as $k \sqrt{\frac{F l}{m}} = v$ .

$k \sqrt{\frac{F l}{m}} = v$

Step 2

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

${(k \sqrt{\frac{F l}{m}})}^{2} = v^{2}$

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 $\sqrt{\frac{F l}{m}}$ as ${(\frac{F l}{m})}^{\frac{1}{2}}$ .

${(k {(\frac{F l}{m})}^{\frac{1}{2}})}^{2} = v^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(k {(\frac{F l}{m})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

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

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Step 3.2.1.1.1

Apply the product rule to $\frac{F l}{m}$ .

${(k \frac{{(F l)}^{\frac{1}{2}}}{m^{\frac{1}{2}}})}^{2} = v^{2}$

Step 3.2.1.1.2

Apply the product rule to $F l$ .

${(k \frac{F^{\frac{1}{2}} l^{\frac{1}{2}}}{m^{\frac{1}{2}}})}^{2} = v^{2}$

Step 3.2.1.2

Combine $k$ and $\frac{F^{\frac{1}{2}} l^{\frac{1}{2}}}{m^{\frac{1}{2}}}$ .

${(\frac{k (F^{\frac{1}{2}} l^{\frac{1}{2}})}{m^{\frac{1}{2}}})}^{2} = v^{2}$

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 $\frac{k F^{\frac{1}{2}} l^{\frac{1}{2}}}{m^{\frac{1}{2}}}$ .

$\frac{{(k F^{\frac{1}{2}} l^{\frac{1}{2}})}^{2}}{{(m^{\frac{1}{2}})}^{2}} = v^{2}$

Step 3.2.1.3.2

Apply the product rule to $k F^{\frac{1}{2}} l^{\frac{1}{2}}$ .

$\frac{{(k F^{\frac{1}{2}})}^{2} {(l^{\frac{1}{2}})}^{2}}{{(m^{\frac{1}{2}})}^{2}} = v^{2}$

Step 3.2.1.3.3

Apply the product rule to $k F^{\frac{1}{2}}$ .

$\frac{k^{2} {(F^{\frac{1}{2}})}^{2} {(l^{\frac{1}{2}})}^{2}}{{(m^{\frac{1}{2}})}^{2}} = v^{2}$

Step 3.2.1.4

Simplify the numerator.

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Step 3.2.1.4.1

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

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Step 3.2.1.4.1.1

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

$\frac{k^{2} F^{\frac{1}{2} \cdot 2} {(l^{\frac{1}{2}})}^{2}}{{(m^{\frac{1}{2}})}^{2}} = v^{2}$

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.

$\frac{k^{2} F^{\frac{1}{2} \cdot 2} {(l^{\frac{1}{2}})}^{2}}{{(m^{\frac{1}{2}})}^{2}} = v^{2}$

Step 3.2.1.4.1.2.2

Rewrite the expression.

$\frac{k^{2} F^{1} {(l^{\frac{1}{2}})}^{2}}{{(m^{\frac{1}{2}})}^{2}} = v^{2}$

Step 3.2.1.4.2

Simplify.

$\frac{k^{2} F {(l^{\frac{1}{2}})}^{2}}{{(m^{\frac{1}{2}})}^{2}} = v^{2}$

Step 3.2.1.4.3

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

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Step 3.2.1.4.3.1

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

$\frac{k^{2} F l^{\frac{1}{2} \cdot 2}}{{(m^{\frac{1}{2}})}^{2}} = v^{2}$

Step 3.2.1.4.3.2

Cancel the common factor of $2$ .

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Step 3.2.1.4.3.2.1

Cancel the common factor.

$\frac{k^{2} F l^{\frac{1}{2} \cdot 2}}{{(m^{\frac{1}{2}})}^{2}} = v^{2}$

Step 3.2.1.4.3.2.2

Rewrite the expression.

$\frac{k^{2} F l^{1}}{{(m^{\frac{1}{2}})}^{2}} = v^{2}$

Step 3.2.1.4.4

Simplify.

$\frac{k^{2} F l}{{(m^{\frac{1}{2}})}^{2}} = v^{2}$

Step 3.2.1.5

Simplify the denominator.

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Step 3.2.1.5.1

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

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Step 3.2.1.5.1.1

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

$\frac{k^{2} F l}{m^{\frac{1}{2} \cdot 2}} = v^{2}$

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.

$\frac{k^{2} F l}{m^{\frac{1}{2} \cdot 2}} = v^{2}$

Step 3.2.1.5.1.2.2

Rewrite the expression.

$\frac{k^{2} F l}{m^{1}} = v^{2}$

Step 3.2.1.5.2

Simplify.

$\frac{k^{2} F l}{m} = v^{2}$

Step 4

Solve for $m$ .

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Step 4.1

Find the LCD of the terms in the equation.

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Step 4.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$m, 1$

Step 4.1.2

The LCM of one and any expression is the expression.

$m$

Step 4.2

Multiply each term in $\frac{k^{2} F l}{m} = v^{2}$ by $m$ to eliminate the fractions.

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Step 4.2.1

Multiply each term in $\frac{k^{2} F l}{m} = v^{2}$ by $m$ .

$\frac{k^{2} F l}{m} m = v^{2} m$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $m$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{k^{2} F l}{m} m = v^{2} m$

Step 4.2.2.1.2

Rewrite the expression.

$k^{2} F l = v^{2} m$

Step 4.3

Solve the equation.

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Step 4.3.1

Rewrite the equation as $v^{2} m = k^{2} F l$ .

$v^{2} m = k^{2} F l$

Step 4.3.2

Divide each term in $v^{2} m = k^{2} F l$ by $v^{2}$ and simplify.

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Step 4.3.2.1

Divide each term in $v^{2} m = k^{2} F l$ by $v^{2}$ .

$\frac{v^{2} m}{v^{2}} = \frac{k^{2} F l}{v^{2}}$

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

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

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Step 4.3.2.2.1.1

Cancel the common factor.

$\frac{v^{2} m}{v^{2}} = \frac{k^{2} F l}{v^{2}}$

Step 4.3.2.2.1.2

Divide $m$ by $1$ .

$m = \frac{k^{2} F l}{v^{2}}$