Solve for x square root of 11x+5- square root of 11x=1

$\sqrt{11 x + 5} - \sqrt{11 x} = 1$

Add $\sqrt{11 x}$ to both sides of the equation.

$\sqrt{11 x + 5} = 1 + \sqrt{11 x}$

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

${\sqrt{11 x + 5}}^{2} = {(1 + \sqrt{11 x})}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11 x + 5}$ as ${(11 x + 5)}^{\frac{1}{2}}$ .

${({(11 x + 5)}^{\frac{1}{2}})}^{2} = {(1 + \sqrt{11 x})}^{2}$

Simplify the left side.

Simplify ${({(11 x + 5)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(11 x + 5)}^{\frac{1}{2}})}^{2}$ .

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

${(11 x + 5)}^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{11 x})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(11 x + 5)}^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{11 x})}^{2}$

Rewrite the expression.

${(11 x + 5)}^{1} = {(1 + \sqrt{11 x})}^{2}$

Simplify.

$11 x + 5 = {(1 + \sqrt{11 x})}^{2}$

Simplify the right side.

Simplify ${(1 + \sqrt{11 x})}^{2}$ .

Rewrite ${(1 + \sqrt{11 x})}^{2}$ as $(1 + \sqrt{11 x}) (1 + \sqrt{11 x})$ .

$11 x + 5 = (1 + \sqrt{11 x}) (1 + \sqrt{11 x})$

Expand $(1 + \sqrt{11 x}) (1 + \sqrt{11 x})$ using the FOIL Method.

Apply the distributive property.

$11 x + 5 = 1 (1 + \sqrt{11 x}) + \sqrt{11 x} (1 + \sqrt{11 x})$

Apply the distributive property.

$11 x + 5 = 1 \cdot 1 + 1 \sqrt{11 x} + \sqrt{11 x} (1 + \sqrt{11 x})$

Apply the distributive property.

$11 x + 5 = 1 \cdot 1 + 1 \sqrt{11 x} + \sqrt{11 x} \cdot 1 + \sqrt{11 x} \sqrt{11 x}$

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$11 x + 5 = 1 + 1 \sqrt{11 x} + \sqrt{11 x} \cdot 1 + \sqrt{11 x} \sqrt{11 x}$

Multiply $\sqrt{11 x}$ by $1$ .

$11 x + 5 = 1 + \sqrt{11 x} + \sqrt{11 x} \cdot 1 + \sqrt{11 x} \sqrt{11 x}$

Multiply $\sqrt{11 x}$ by $1$ .

$11 x + 5 = 1 + \sqrt{11 x} + \sqrt{11 x} + \sqrt{11 x} \sqrt{11 x}$

Multiply $\sqrt{11 x} \sqrt{11 x}$ .

Raise $\sqrt{11 x}$ to the power of $1$ .

$11 x + 5 = 1 + \sqrt{11 x} + \sqrt{11 x} + {\sqrt{11 x}}^{1} \sqrt{11 x}$

Raise $\sqrt{11 x}$ to the power of $1$ .

$11 x + 5 = 1 + \sqrt{11 x} + \sqrt{11 x} + {\sqrt{11 x}}^{1} {\sqrt{11 x}}^{1}$

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

$11 x + 5 = 1 + \sqrt{11 x} + \sqrt{11 x} + {\sqrt{11 x}}^{1 + 1}$

Add $1$ and $1$ .

$11 x + 5 = 1 + \sqrt{11 x} + \sqrt{11 x} + {\sqrt{11 x}}^{2}$

Rewrite ${\sqrt{11 x}}^{2}$ as $11 x$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11 x}$ as ${(11 x)}^{\frac{1}{2}}$ .

$11 x + 5 = 1 + \sqrt{11 x} + \sqrt{11 x} + {({(11 x)}^{\frac{1}{2}})}^{2}$

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

$11 x + 5 = 1 + \sqrt{11 x} + \sqrt{11 x} + {(11 x)}^{\frac{1}{2} \cdot 2}$

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

$11 x + 5 = 1 + \sqrt{11 x} + \sqrt{11 x} + {(11 x)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$11 x + 5 = 1 + \sqrt{11 x} + \sqrt{11 x} + {(11 x)}^{\frac{2}{2}}$

Rewrite the expression.

$11 x + 5 = 1 + \sqrt{11 x} + \sqrt{11 x} + {(11 x)}^{1}$

Simplify.

$11 x + 5 = 1 + \sqrt{11 x} + \sqrt{11 x} + 11 x$

Add $\sqrt{11 x}$ and $\sqrt{11 x}$ .

$11 x + 5 = 1 + 2 \sqrt{11 x} + 11 x$

Solve for $2 \sqrt{11 x}$ .

Rewrite the equation as $1 + 2 \sqrt{11 x} + 11 x = 11 x + 5$ .

$1 + 2 \sqrt{11 x} + 11 x = 11 x + 5$

Move all terms not containing $2 \sqrt{11 x}$ to the right side of the equation.

Subtract $1$ from both sides of the equation.

$2 \sqrt{11 x} + 11 x = 11 x + 5 - 1$

Subtract $11 x$ from both sides of the equation.

$2 \sqrt{11 x} = 11 x + 5 - 1 - 11 x$

Combine the opposite terms in $11 x + 5 - 1 - 11 x$ .

Subtract $11 x$ from $11 x$ .

$2 \sqrt{11 x} = 0 + 5 - 1$

Add $0$ and $5$ .

$2 \sqrt{11 x} = 5 - 1$

Subtract $1$ from $5$ .

$2 \sqrt{11 x} = 4$

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

${(2 \sqrt{11 x})}^{2} = 4^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11 x}$ as ${(11 x)}^{\frac{1}{2}}$ .

${(2 {(11 x)}^{\frac{1}{2}})}^{2} = 4^{2}$

Simplify the left side.

Simplify ${(2 {(11 x)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $11 x$ .

${(2 (11^{\frac{1}{2}} x^{\frac{1}{2}}))}^{2} = 4^{2}$

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

Apply the product rule to $2 \cdot 11^{\frac{1}{2}} x^{\frac{1}{2}}$ .

${(2 \cdot 11^{\frac{1}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = 4^{2}$

Apply the product rule to $2 \cdot 11^{\frac{1}{2}}$ .

$2^{2} \cdot {(11^{\frac{1}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = 4^{2}$

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

$4 \cdot {(11^{\frac{1}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = 4^{2}$

Multiply the exponents in ${(11^{\frac{1}{2}})}^{2}$ .

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

$4 \cdot 11^{\frac{1}{2} \cdot 2} {(x^{\frac{1}{2}})}^{2} = 4^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 \cdot 11^{\frac{1}{2} \cdot 2} {(x^{\frac{1}{2}})}^{2} = 4^{2}$

Rewrite the expression.

$4 \cdot 11^{1} {(x^{\frac{1}{2}})}^{2} = 4^{2}$

Evaluate the exponent.

$4 \cdot 11 {(x^{\frac{1}{2}})}^{2} = 4^{2}$

Multiply $4$ by $11$ .

$44 {(x^{\frac{1}{2}})}^{2} = 4^{2}$

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

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

$44 x^{\frac{1}{2} \cdot 2} = 4^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$44 x^{\frac{1}{2} \cdot 2} = 4^{2}$

Rewrite the expression.

$44 x^{1} = 4^{2}$

Simplify.

$44 x = 4^{2}$

Simplify the right side.

Raise $4$ to the power of $2$ .

$44 x = 16$

Divide each term in $44 x = 16$ by $44$ and simplify.

Divide each term in $44 x = 16$ by $44$ .

$\frac{44 x}{44} = \frac{16}{44}$

Simplify the left side.

Cancel the common factor of $44$ .

Cancel the common factor.

$\frac{44 x}{44} = \frac{16}{44}$

Divide $x$ by $1$ .

$x = \frac{16}{44}$

Simplify the right side.

Cancel the common factor of $16$ and $44$ .

Factor $4$ out of $16$ .

$x = \frac{4 (4)}{44}$

Cancel the common factors.

Factor $4$ out of $44$ .

$x = \frac{4 \cdot 4}{4 \cdot 11}$

Cancel the common factor.

$x = \frac{4 \cdot 4}{4 \cdot 11}$

Rewrite the expression.

$x = \frac{4}{11}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4}{11}$

Decimal Form:

$x = 0.\overline{36}$

Solve for x square root of 11x+5- square root of 11x=1

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