Solve for v -10 square root of v/8=-40

$- 10 \sqrt{\frac{v}{8}} = - 40$

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

${(- 10 \sqrt{\frac{v}{8}})}^{2} = {(- 40)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{v}{8}}$ as ${(\frac{v}{8})}^{\frac{1}{2}}$ .

${(- 10 {(\frac{v}{8})}^{\frac{1}{2}})}^{2} = {(- 40)}^{2}$

Simplify the left side.

Simplify ${(- 10 {(\frac{v}{8})}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $\frac{v}{8}$ .

${(- 10 \frac{v^{\frac{1}{2}}}{8^{\frac{1}{2}}})}^{2} = {(- 40)}^{2}$

Combine $- 10$ and $\frac{v^{\frac{1}{2}}}{8^{\frac{1}{2}}}$ .

${(\frac{- 10 v^{\frac{1}{2}}}{8^{\frac{1}{2}}})}^{2} = {(- 40)}^{2}$

Move the negative in front of the fraction.

${(- \frac{(10) v^{\frac{1}{2}}}{8^{\frac{1}{2}}})}^{2} = {(- 40)}^{2}$

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

Apply the product rule to $- \frac{10 v^{\frac{1}{2}}}{8^{\frac{1}{2}}}$ .

${(- 1)}^{2} {(\frac{10 v^{\frac{1}{2}}}{8^{\frac{1}{2}}})}^{2} = {(- 40)}^{2}$

Apply the product rule to $\frac{10 v^{\frac{1}{2}}}{8^{\frac{1}{2}}}$ .

${(- 1)}^{2} \frac{{(10 v^{\frac{1}{2}})}^{2}}{{(8^{\frac{1}{2}})}^{2}} = {(- 40)}^{2}$

Apply the product rule to $10 v^{\frac{1}{2}}$ .

${(- 1)}^{2} \frac{10^{2} {(v^{\frac{1}{2}})}^{2}}{{(8^{\frac{1}{2}})}^{2}} = {(- 40)}^{2}$

Simplify the expression.

Raise $- 1$ to the power of $2$ .

$1 \frac{10^{2} {(v^{\frac{1}{2}})}^{2}}{{(8^{\frac{1}{2}})}^{2}} = {(- 40)}^{2}$

Multiply $\frac{10^{2} {(v^{\frac{1}{2}})}^{2}}{{(8^{\frac{1}{2}})}^{2}}$ by $1$ .

$\frac{10^{2} {(v^{\frac{1}{2}})}^{2}}{{(8^{\frac{1}{2}})}^{2}} = {(- 40)}^{2}$

Simplify the numerator.

Raise $10$ to the power of $2$ .

$\frac{100 {(v^{\frac{1}{2}})}^{2}}{{(8^{\frac{1}{2}})}^{2}} = {(- 40)}^{2}$

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

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

$\frac{100 v^{\frac{1}{2} \cdot 2}}{{(8^{\frac{1}{2}})}^{2}} = {(- 40)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{100 v^{\frac{1}{2} \cdot 2}}{{(8^{\frac{1}{2}})}^{2}} = {(- 40)}^{2}$

Rewrite the expression.

$\frac{100 v^{1}}{{(8^{\frac{1}{2}})}^{2}} = {(- 40)}^{2}$

Simplify.

$\frac{100 v}{{(8^{\frac{1}{2}})}^{2}} = {(- 40)}^{2}$

Simplify the denominator.

Multiply the exponents in ${(8^{\frac{1}{2}})}^{2}$ .

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

$\frac{100 v}{8^{\frac{1}{2} \cdot 2}} = {(- 40)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{100 v}{8^{\frac{1}{2} \cdot 2}} = {(- 40)}^{2}$

Rewrite the expression.

$\frac{100 v}{8^{1}} = {(- 40)}^{2}$

Evaluate the exponent.

$\frac{100 v}{8} = {(- 40)}^{2}$

Cancel the common factor of $100$ and $8$ .

Factor $4$ out of $100 v$ .

$\frac{4 (25 v)}{8} = {(- 40)}^{2}$

Cancel the common factors.

Factor $4$ out of $8$ .

$\frac{4 (25 v)}{4 (2)} = {(- 40)}^{2}$

Cancel the common factor.

$\frac{4 (25 v)}{4 \cdot 2} = {(- 40)}^{2}$

Rewrite the expression.

$\frac{25 v}{2} = {(- 40)}^{2}$

Simplify the right side.

Raise $- 40$ to the power of $2$ .

$\frac{25 v}{2} = 1600$

Solve for $v$ .

Multiply both sides of the equation by $\frac{2}{25}$ .

$\frac{2}{25} \cdot \frac{25 v}{2} = \frac{2}{25} \cdot 1600$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{2}{25} \cdot \frac{25 v}{2}$ .

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2}{25} \cdot \frac{25 v}{2} = \frac{2}{25} \cdot 1600$

Rewrite the expression.

$\frac{1}{25} (25 v) = \frac{2}{25} \cdot 1600$

Cancel the common factor of $25$ .

Factor $25$ out of $25 v$ .

$\frac{1}{25} (25 (v)) = \frac{2}{25} \cdot 1600$

Cancel the common factor.

$\frac{1}{25} (25 v) = \frac{2}{25} \cdot 1600$

Rewrite the expression.

$v = \frac{2}{25} \cdot 1600$

Simplify the right side.

Simplify $\frac{2}{25} \cdot 1600$ .

Cancel the common factor of $25$ .

Factor $25$ out of $1600$ .

$v = \frac{2}{25} \cdot (25 (64))$

Cancel the common factor.

$v = \frac{2}{25} \cdot (25 \cdot 64)$

Rewrite the expression.

$v = 2 \cdot 64$

Multiply $2$ by $64$ .

$v = 128$

Solve for v -10 square root of v/8=-40

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