Solve for z square root of z-7 = square root of z-105

$\sqrt{z} - 7 = \sqrt{z - 105}$

Step 1

Add $7$ to both sides of the equation.

$\sqrt{z} = \sqrt{z - 105} + 7$

Step 2

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

${\sqrt{z}}^{2} = {(\sqrt{z - 105} + 7)}^{2}$

Step 3

Simplify each side of the equation.

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

${(z^{\frac{1}{2}})}^{2} = {(\sqrt{z - 105} + 7)}^{2}$

Simplify the left side.

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

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

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

$z^{\frac{1}{2} \cdot 2} = {(\sqrt{z - 105} + 7)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$z^{\frac{1}{2} \cdot 2} = {(\sqrt{z - 105} + 7)}^{2}$

Rewrite the expression.

$z^{1} = {(\sqrt{z - 105} + 7)}^{2}$

Simplify.

$z = {(\sqrt{z - 105} + 7)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{z - 105} + 7)}^{2}$ .

Rewrite ${(\sqrt{z - 105} + 7)}^{2}$ as $(\sqrt{z - 105} + 7) (\sqrt{z - 105} + 7)$ .

$z = (\sqrt{z - 105} + 7) (\sqrt{z - 105} + 7)$

Expand $(\sqrt{z - 105} + 7) (\sqrt{z - 105} + 7)$ using the FOIL Method.

Apply the distributive property.

$z = \sqrt{z - 105} (\sqrt{z - 105} + 7) + 7 (\sqrt{z - 105} + 7)$

Apply the distributive property.

$z = \sqrt{z - 105} \sqrt{z - 105} + \sqrt{z - 105} \cdot 7 + 7 (\sqrt{z - 105} + 7)$

Apply the distributive property.

$z = \sqrt{z - 105} \sqrt{z - 105} + \sqrt{z - 105} \cdot 7 + 7 \sqrt{z - 105} + 7 \cdot 7$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{z - 105} \sqrt{z - 105}$ .

Raise $\sqrt{z - 105}$ to the power of $1$ .

$z = {\sqrt{z - 105}}^{1} \sqrt{z - 105} + \sqrt{z - 105} \cdot 7 + 7 \sqrt{z - 105} + 7 \cdot 7$

Raise $\sqrt{z - 105}$ to the power of $1$ .

$z = {\sqrt{z - 105}}^{1} {\sqrt{z - 105}}^{1} + \sqrt{z - 105} \cdot 7 + 7 \sqrt{z - 105} + 7 \cdot 7$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$z = {\sqrt{z - 105}}^{1 + 1} + \sqrt{z - 105} \cdot 7 + 7 \sqrt{z - 105} + 7 \cdot 7$

Add $1$ and $1$ .

$z = {\sqrt{z - 105}}^{2} + \sqrt{z - 105} \cdot 7 + 7 \sqrt{z - 105} + 7 \cdot 7$

Rewrite ${\sqrt{z - 105}}^{2}$ as $z - 105$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z - 105}$ as ${(z - 105)}^{\frac{1}{2}}$ .

$z = {({(z - 105)}^{\frac{1}{2}})}^{2} + \sqrt{z - 105} \cdot 7 + 7 \sqrt{z - 105} + 7 \cdot 7$

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

$z = {(z - 105)}^{\frac{1}{2} \cdot 2} + \sqrt{z - 105} \cdot 7 + 7 \sqrt{z - 105} + 7 \cdot 7$

Combine $\frac{1}{2}$ and $2$ .

$z = {(z - 105)}^{\frac{2}{2}} + \sqrt{z - 105} \cdot 7 + 7 \sqrt{z - 105} + 7 \cdot 7$

Cancel the common factor of $2$ .

Cancel the common factor.

$z = {(z - 105)}^{\frac{2}{2}} + \sqrt{z - 105} \cdot 7 + 7 \sqrt{z - 105} + 7 \cdot 7$

Rewrite the expression.

$z = {(z - 105)}^{1} + \sqrt{z - 105} \cdot 7 + 7 \sqrt{z - 105} + 7 \cdot 7$

Simplify.

$z = z - 105 + \sqrt{z - 105} \cdot 7 + 7 \sqrt{z - 105} + 7 \cdot 7$

Move $7$ to the left of $\sqrt{z - 105}$ .

$z = z - 105 + 7 \cdot \sqrt{z - 105} + 7 \sqrt{z - 105} + 7 \cdot 7$

Multiply $7$ by $7$ .

$z = z - 105 + 7 \sqrt{z - 105} + 7 \sqrt{z - 105} + 49$

Add $- 105$ and $49$ .

$z = z - 56 + 7 \sqrt{z - 105} + 7 \sqrt{z - 105}$

Add $7 \sqrt{z - 105}$ and $7 \sqrt{z - 105}$ .

$z = z - 56 + 14 \sqrt{z - 105}$

Step 4

Solve for $14 \sqrt{z - 105}$ .

Rewrite the equation as $z - 56 + 14 \sqrt{z - 105} = z$ .

$z - 56 + 14 \sqrt{z - 105} = z$

Move all terms not containing $14 \sqrt{z - 105}$ to the right side of the equation.

Subtract $z$ from both sides of the equation.

$- 56 + 14 \sqrt{z - 105} = z - z$

Add $56$ to both sides of the equation.

$14 \sqrt{z - 105} = z - z + 56$

Subtract $z$ from $z$ .

$14 \sqrt{z - 105} = 0 + 56$

Add $0$ and $56$ .

$14 \sqrt{z - 105} = 56$

Step 5

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

${(14 \sqrt{z - 105})}^{2} = 56^{2}$

Step 6

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z - 105}$ as ${(z - 105)}^{\frac{1}{2}}$ .

${(14 {(z - 105)}^{\frac{1}{2}})}^{2} = 56^{2}$

Simplify the left side.

Simplify ${(14 {(z - 105)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $14 {(z - 105)}^{\frac{1}{2}}$ .

$14^{2} {({(z - 105)}^{\frac{1}{2}})}^{2} = 56^{2}$

Raise $14$ to the power of $2$ .

$196 {({(z - 105)}^{\frac{1}{2}})}^{2} = 56^{2}$

Multiply the exponents in ${({(z - 105)}^{\frac{1}{2}})}^{2}$ .

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

$196 {(z - 105)}^{\frac{1}{2} \cdot 2} = 56^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$196 {(z - 105)}^{\frac{1}{2} \cdot 2} = 56^{2}$

Rewrite the expression.

$196 {(z - 105)}^{1} = 56^{2}$

Simplify.

$196 (z - 105) = 56^{2}$

Apply the distributive property.

$196 z + 196 \cdot - 105 = 56^{2}$

Multiply $196$ by $- 105$ .

$196 z - 20580 = 56^{2}$

Simplify the right side.

Raise $56$ to the power of $2$ .

$196 z - 20580 = 3136$

Step 7

Solve for $z$ .

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

Add $20580$ to both sides of the equation.

$196 z = 3136 + 20580$

Add $3136$ and $20580$ .

$196 z = 23716$

Divide each term in $196 z = 23716$ by $196$ and simplify.

Divide each term in $196 z = 23716$ by $196$ .

$\frac{196 z}{196} = \frac{23716}{196}$

Simplify the left side.

Cancel the common factor of $196$ .

Cancel the common factor.

$\frac{196 z}{196} = \frac{23716}{196}$

Divide $z$ by $1$ .

$z = \frac{23716}{196}$

Simplify the right side.

Divide $23716$ by $196$ .

$z = 121$

Solve for z square root of z-7 = square root of z-105

Related Questions

Support LCMGCF.com by disabling your adblocker.