Solve for z square root of 18/(2z-52)=1/8

$\sqrt{\frac{18}{2 z - 52}} = \frac{1}{8}$

Step 1

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

${\sqrt{\frac{18}{2 z - 52}}}^{2} = {(\frac{1}{8})}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{18}{2 z - 52}}$ as ${(\frac{18}{2 z - 52})}^{\frac{1}{2}}$ .

${({(\frac{18}{2 z - 52})}^{\frac{1}{2}})}^{2} = {(\frac{1}{8})}^{2}$

Simplify the left side.

Simplify ${({(\frac{18}{2 z - 52})}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(\frac{18}{2 z - 52})}^{\frac{1}{2}})}^{2}$ .

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

${(\frac{18}{2 z - 52})}^{\frac{1}{2} \cdot 2} = {(\frac{1}{8})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(\frac{18}{2 z - 52})}^{\frac{1}{2} \cdot 2} = {(\frac{1}{8})}^{2}$

Rewrite the expression.

${(\frac{18}{2 z - 52})}^{1} = {(\frac{1}{8})}^{2}$

Cancel the common factor of $18$ and $2 z - 52$ .

Factor $2$ out of $18$ .

${(\frac{2 \cdot 9}{2 z - 52})}^{1} = {(\frac{1}{8})}^{2}$

Cancel the common factors.

Factor $2$ out of $2 z$ .

${(\frac{2 \cdot 9}{2 (z) - 52})}^{1} = {(\frac{1}{8})}^{2}$

Factor $2$ out of $- 52$ .

${(\frac{2 \cdot 9}{2 (z) + 2 (- 26)})}^{1} = {(\frac{1}{8})}^{2}$

Factor $2$ out of $2 (z) + 2 (- 26)$ .

${(\frac{2 \cdot 9}{2 (z - 26)})}^{1} = {(\frac{1}{8})}^{2}$

Cancel the common factor.

${(\frac{2 \cdot 9}{2 (z - 26)})}^{1} = {(\frac{1}{8})}^{2}$

Rewrite the expression.

${(\frac{9}{z - 26})}^{1} = {(\frac{1}{8})}^{2}$

Simplify.

$\frac{9}{z - 26} = {(\frac{1}{8})}^{2}$

Simplify the right side.

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

Apply the product rule to $\frac{1}{8}$ .

$\frac{9}{z - 26} = \frac{1^{2}}{8^{2}}$

One to any power is one.

$\frac{9}{z - 26} = \frac{1}{8^{2}}$

Raise $8$ to the power of $2$ .

$\frac{9}{z - 26} = \frac{1}{64}$

Step 3

Solve for $z$ .

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$9 \cdot 64 = (z - 26) \cdot 1$

Solve the equation for $z$ .

Rewrite the equation as $(z - 26) \cdot 1 = 9 \cdot 64$ .

$(z - 26) \cdot 1 = 9 \cdot 64$

Multiply $z - 26$ by $1$ .

$z - 26 = 9 \cdot 64$

Multiply $9$ by $64$ .

$z - 26 = 576$

Move all terms not containing $z$ to the right side of the equation.

Add $26$ to both sides of the equation.

$z = 576 + 26$

Add $576$ and $26$ .

$z = 602$

Solve for z square root of 18/(2z-52)=1/8

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