Solve for x square root of 5x-7 = square root of x-10

$\sqrt{5 x - 7} = \sqrt{x - 10}$

Step 1

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

${\sqrt{5 x - 7}}^{2} = {\sqrt{x - 10}}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x - 7}$ as ${(5 x - 7)}^{\frac{1}{2}}$ .

${({(5 x - 7)}^{\frac{1}{2}})}^{2} = {\sqrt{x - 10}}^{2}$

Simplify the left side.

Simplify ${({(5 x - 7)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(5 x - 7)}^{\frac{1}{2}})}^{2}$ .

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

${(5 x - 7)}^{\frac{1}{2} \cdot 2} = {\sqrt{x - 10}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(5 x - 7)}^{\frac{1}{2} \cdot 2} = {\sqrt{x - 10}}^{2}$

Rewrite the expression.

${(5 x - 7)}^{1} = {\sqrt{x - 10}}^{2}$

Simplify.

$5 x - 7 = {\sqrt{x - 10}}^{2}$

Simplify the right side.

Rewrite ${\sqrt{x - 10}}^{2}$ as $x - 10$ .

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

$5 x - 7 = {({(x - 10)}^{\frac{1}{2}})}^{2}$

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

$5 x - 7 = {(x - 10)}^{\frac{1}{2} \cdot 2}$

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

$5 x - 7 = {(x - 10)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$5 x - 7 = {(x - 10)}^{\frac{2}{2}}$

Rewrite the expression.

$5 x - 7 = {(x - 10)}^{1}$

Simplify.

$5 x - 7 = x - 10$

Step 3

Solve for $x$ .

Move all terms containing $x$ to the left side of the equation.

Subtract $x$ from both sides of the equation.

$5 x - 7 - x = - 10$

Subtract $x$ from $5 x$ .

$4 x - 7 = - 10$

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

Add $7$ to both sides of the equation.

$4 x = - 10 + 7$

Add $- 10$ and $7$ .

$4 x = - 3$

Divide each term in $4 x = - 3$ by $4$ and simplify.

Divide each term in $4 x = - 3$ by $4$ .

$\frac{4 x}{4} = \frac{- 3}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 3}{4}$

Divide $x$ by $1$ .

$x = \frac{- 3}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{3}{4}$

Step 4

Exclude the solutions that do not make $\sqrt{5 x - 7} = \sqrt{x - 10}$ true.

$No solution$

Solve for x square root of 5x-7 = square root of x-10

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