Solve for a square root of 9a-5- square root of 5a+8=0

$\sqrt{9 a - 5} - \sqrt{5 a + 8} = 0$

Step 1

Add $\sqrt{5 a + 8}$ to both sides of the equation.

$\sqrt{9 a - 5} = \sqrt{5 a + 8}$

Step 2

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

${\sqrt{9 a - 5}}^{2} = {\sqrt{5 a + 8}}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 a - 5}$ as ${(9 a - 5)}^{\frac{1}{2}}$ .

${({(9 a - 5)}^{\frac{1}{2}})}^{2} = {\sqrt{5 a + 8}}^{2}$

Simplify the left side.

Simplify ${({(9 a - 5)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(9 a - 5)}^{\frac{1}{2}})}^{2}$ .

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

${(9 a - 5)}^{\frac{1}{2} \cdot 2} = {\sqrt{5 a + 8}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(9 a - 5)}^{\frac{1}{2} \cdot 2} = {\sqrt{5 a + 8}}^{2}$

Rewrite the expression.

${(9 a - 5)}^{1} = {\sqrt{5 a + 8}}^{2}$

Simplify.

$9 a - 5 = {\sqrt{5 a + 8}}^{2}$

Simplify the right side.

Rewrite ${\sqrt{5 a + 8}}^{2}$ as $5 a + 8$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 a + 8}$ as ${(5 a + 8)}^{\frac{1}{2}}$ .

$9 a - 5 = {({(5 a + 8)}^{\frac{1}{2}})}^{2}$

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

$9 a - 5 = {(5 a + 8)}^{\frac{1}{2} \cdot 2}$

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

$9 a - 5 = {(5 a + 8)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$9 a - 5 = {(5 a + 8)}^{\frac{2}{2}}$

Rewrite the expression.

$9 a - 5 = {(5 a + 8)}^{1}$

Simplify.

$9 a - 5 = 5 a + 8$

Step 4

Solve for $a$ .

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

Subtract $5 a$ from both sides of the equation.

$9 a - 5 - 5 a = 8$

Subtract $5 a$ from $9 a$ .

$4 a - 5 = 8$

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

Add $5$ to both sides of the equation.

$4 a = 8 + 5$

Add $8$ and $5$ .

$4 a = 13$

Divide each term in $4 a = 13$ by $4$ and simplify.

Divide each term in $4 a = 13$ by $4$ .

$\frac{4 a}{4} = \frac{13}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 a}{4} = \frac{13}{4}$

Divide $a$ by $1$ .

$a = \frac{13}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$a = \frac{13}{4}$

Decimal Form:

$a = 3.25$

Mixed Number Form:

$a = 3 \frac{1}{4}$

Solve for a square root of 9a-5- square root of 5a+8=0

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