Precalculus Examples

$m = \sqrt{5 m}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{5 m} = m$

Step 2

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

${\sqrt{5 m}}^{2} = m^{2}$

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 $\sqrt{5 m}$ as ${(5 m)}^{\frac{1}{2}}$ .

${({(5 m)}^{\frac{1}{2}})}^{2} = m^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(5 m)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(5 m)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(5 m)}^{\frac{1}{2} \cdot 2} = m^{2}$

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.

${(5 m)}^{\frac{1}{2} \cdot 2} = m^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(5 m)}^{1} = m^{2}$

Step 3.2.1.2

Simplify.

$5 m = m^{2}$

Step 4

Solve for $m$ .

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Step 4.1

Subtract $m^{2}$ from both sides of the equation.

$5 m - m^{2} = 0$

Step 4.2

Factor the left side of the equation.

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Step 4.2.1

Let $u = m$ . Substitute $u$ for all occurrences of $m$ .

$5 u - u^{2} = 0$

Step 4.2.2

Factor $u$ out of $5 u - u^{2}$ .

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Step 4.2.2.1

Factor $u$ out of $5 u$ .

$u \cdot 5 - u^{2} = 0$

Step 4.2.2.2

Factor $u$ out of $- u^{2}$ .

$u \cdot 5 + u (- u) = 0$

Step 4.2.2.3

Factor $u$ out of $u \cdot 5 + u (- u)$ .

$u (5 - u) = 0$

Step 4.2.3

Replace all occurrences of $u$ with $m$ .

$m (5 - m) = 0$

Step 4.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$m = 0$

$5 - m = 0$

Step 4.4

Set $m$ equal to $0$ .

$m = 0$

Step 4.5

Set $5 - m$ equal to $0$ and solve for $m$ .

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Step 4.5.1

Set $5 - m$ equal to $0$ .

$5 - m = 0$

Step 4.5.2

Solve $5 - m = 0$ for $m$ .

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Step 4.5.2.1

Subtract $5$ from both sides of the equation.

$- m = - 5$

Step 4.5.2.2

Divide each term in $- m = - 5$ by $- 1$ and simplify.

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Step 4.5.2.2.1

Divide each term in $- m = - 5$ by $- 1$ .

$\frac{- m}{- 1} = \frac{- 5}{- 1}$

Step 4.5.2.2.2

Simplify the left side.

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Step 4.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{m}{1} = \frac{- 5}{- 1}$

Step 4.5.2.2.2.2

Divide $m$ by $1$ .

$m = \frac{- 5}{- 1}$

Step 4.5.2.2.3

Simplify the right side.

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Step 4.5.2.2.3.1

Divide $- 5$ by $- 1$ .

$m = 5$

Step 4.6

The final solution is all the values that make $m (5 - m) = 0$ true.

$m = 0, 5$