Solve for r r = square root of 10r

$r = \sqrt{10 r}$

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.

$\sqrt{10 r} = r$

Step 2

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

${\sqrt{10 r}}^{2} = r^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10 r}$ as ${(10 r)}^{\frac{1}{2}}$ .

${({(10 r)}^{\frac{1}{2}})}^{2} = r^{2}$

Simplify the left side.

Simplify ${({(10 r)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(10 r)}^{\frac{1}{2}})}^{2}$ .

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

${(10 r)}^{\frac{1}{2} \cdot 2} = r^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(10 r)}^{\frac{1}{2} \cdot 2} = r^{2}$

Rewrite the expression.

${(10 r)}^{1} = r^{2}$

Simplify.

$10 r = r^{2}$

Step 4

Solve for $r$ .

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

$10 r - r^{2} = 0$

Factor the left side of the equation.

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

$10 u - u^{2} = 0$

Factor $u$ out of $10 u - u^{2}$ .

Factor $u$ out of $10 u$ .

$u \cdot 10 - u^{2} = 0$

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

$u \cdot 10 + u (- u) = 0$

Factor $u$ out of $u \cdot 10 + u (- u)$ .

$u (10 - u) = 0$

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

$r (10 - r) = 0$

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

$r = 0$

$10 - r = 0$

Set $r$ equal to $0$ .

$r = 0$

Set $10 - r$ equal to $0$ and solve for $r$ .

Set $10 - r$ equal to $0$ .

$10 - r = 0$

Solve $10 - r = 0$ for $r$ .

Subtract $10$ from both sides of the equation.

$- r = - 10$

Divide each term in $- r = - 10$ by $- 1$ and simplify.

Divide each term in $- r = - 10$ by $- 1$ .

$\frac{- r}{- 1} = \frac{- 10}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{r}{1} = \frac{- 10}{- 1}$

Divide $r$ by $1$ .

$r = \frac{- 10}{- 1}$

Simplify the right side.

Divide $- 10$ by $- 1$ .

$r = 10$

The final solution is all the values that make $r (10 - r) = 0$ true.

$r = 0, 10$

Solve for r r = square root of 10r

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