Precalculus Examples

$\sqrt{1 + 18 \sqrt{x}} = \sqrt{x} + 1$

Step 1

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

${\sqrt{1 + 18 \sqrt{x}}}^{2} = {(\sqrt{x} + 1)}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

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

${({(1 + 18 \sqrt{x})}^{\frac{1}{2}})}^{2} = {(\sqrt{x} + 1)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(1 + 18 \sqrt{x})}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(1 + 18 \sqrt{x})}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(1 + 18 \sqrt{x})}^{\frac{1}{2} \cdot 2} = {(\sqrt{x} + 1)}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(1 + 18 \sqrt{x})}^{\frac{1}{2} \cdot 2} = {(\sqrt{x} + 1)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(1 + 18 \sqrt{x})}^{1} = {(\sqrt{x} + 1)}^{2}$

Step 2.2.1.2

Simplify.

$1 + 18 \sqrt{x} = {(\sqrt{x} + 1)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

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

$1 + 18 \sqrt{x} = (\sqrt{x} + 1) (\sqrt{x} + 1)$

Step 2.3.1.2

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

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Step 2.3.1.2.1

Apply the distributive property.

$1 + 18 \sqrt{x} = \sqrt{x} (\sqrt{x} + 1) + 1 (\sqrt{x} + 1)$

Step 2.3.1.2.2

Apply the distributive property.

$1 + 18 \sqrt{x} = \sqrt{x} \sqrt{x} + \sqrt{x} \cdot 1 + 1 (\sqrt{x} + 1)$

Step 2.3.1.2.3

Apply the distributive property.

$1 + 18 \sqrt{x} = \sqrt{x} \sqrt{x} + \sqrt{x} \cdot 1 + 1 \sqrt{x} + 1 \cdot 1$

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

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

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Step 2.3.1.3.1.1.1

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

$1 + 18 \sqrt{x} = {\sqrt{x}}^{1} \sqrt{x} + \sqrt{x} \cdot 1 + 1 \sqrt{x} + 1 \cdot 1$

Step 2.3.1.3.1.1.2

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

$1 + 18 \sqrt{x} = {\sqrt{x}}^{1} {\sqrt{x}}^{1} + \sqrt{x} \cdot 1 + 1 \sqrt{x} + 1 \cdot 1$

Step 2.3.1.3.1.1.3

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

$1 + 18 \sqrt{x} = {\sqrt{x}}^{1 + 1} + \sqrt{x} \cdot 1 + 1 \sqrt{x} + 1 \cdot 1$

Step 2.3.1.3.1.1.4

Add $1$ and $1$ .

$1 + 18 \sqrt{x} = {\sqrt{x}}^{2} + \sqrt{x} \cdot 1 + 1 \sqrt{x} + 1 \cdot 1$

Step 2.3.1.3.1.2

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

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Step 2.3.1.3.1.2.1

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

$1 + 18 \sqrt{x} = {(x^{\frac{1}{2}})}^{2} + \sqrt{x} \cdot 1 + 1 \sqrt{x} + 1 \cdot 1$

Step 2.3.1.3.1.2.2

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

$1 + 18 \sqrt{x} = x^{\frac{1}{2} \cdot 2} + \sqrt{x} \cdot 1 + 1 \sqrt{x} + 1 \cdot 1$

Step 2.3.1.3.1.2.3

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

$1 + 18 \sqrt{x} = x^{\frac{2}{2}} + \sqrt{x} \cdot 1 + 1 \sqrt{x} + 1 \cdot 1$

Step 2.3.1.3.1.2.4

Cancel the common factor of $2$ .

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Step 2.3.1.3.1.2.4.1

Cancel the common factor.

$1 + 18 \sqrt{x} = x^{\frac{2}{2}} + \sqrt{x} \cdot 1 + 1 \sqrt{x} + 1 \cdot 1$

Step 2.3.1.3.1.2.4.2

Rewrite the expression.

$1 + 18 \sqrt{x} = x^{1} + \sqrt{x} \cdot 1 + 1 \sqrt{x} + 1 \cdot 1$

Step 2.3.1.3.1.2.5

Simplify.

$1 + 18 \sqrt{x} = x + \sqrt{x} \cdot 1 + 1 \sqrt{x} + 1 \cdot 1$

Step 2.3.1.3.1.3

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

$1 + 18 \sqrt{x} = x + \sqrt{x} + 1 \sqrt{x} + 1 \cdot 1$

Step 2.3.1.3.1.4

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

$1 + 18 \sqrt{x} = x + \sqrt{x} + \sqrt{x} + 1 \cdot 1$

Step 2.3.1.3.1.5

Multiply $1$ by $1$ .

$1 + 18 \sqrt{x} = x + \sqrt{x} + \sqrt{x} + 1$

Step 2.3.1.3.2

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

$1 + 18 \sqrt{x} = x + 2 \sqrt{x} + 1$

Step 3

Solve for $\sqrt{x}$ .

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Step 3.1

Move all terms containing $\sqrt{x}$ to the left side of the equation.

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Step 3.1.1

Subtract $2 \sqrt{x}$ from both sides of the equation.

$1 + 18 \sqrt{x} - 2 \sqrt{x} = x + 1$

Step 3.1.2

Subtract $2 \sqrt{x}$ from $18 \sqrt{x}$ .

$1 + 16 \sqrt{x} = x + 1$

Step 3.2

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

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Step 3.2.1

Subtract $1$ from both sides of the equation.

$16 \sqrt{x} = x + 1 - 1$

Step 3.2.2

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

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Step 3.2.2.1

Subtract $1$ from $1$ .

$16 \sqrt{x} = x + 0$

Step 3.2.2.2

Add $x$ and $0$ .

$16 \sqrt{x} = x$

Step 3.3

Divide each term in $16 \sqrt{x} = x$ by $16$ and simplify.

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Step 3.3.1

Divide each term in $16 \sqrt{x} = x$ by $16$ .

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $16$ .

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Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Divide $\sqrt{x}$ by $1$ .

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

Step 4

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

${\sqrt{x}}^{2} = {(\frac{x}{16})}^{2}$

Step 5

Simplify each side of the equation.

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Step 5.1

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

${(x^{\frac{1}{2}})}^{2} = {(\frac{x}{16})}^{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

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

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Step 5.2.1.1

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

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Step 5.2.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = {(\frac{x}{16})}^{2}$

Step 5.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(\frac{x}{16})}^{2}$

Step 5.2.1.1.2.2

Rewrite the expression.

$x^{1} = {(\frac{x}{16})}^{2}$

Step 5.2.1.2

Simplify.

$x = {(\frac{x}{16})}^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify ${(\frac{x}{16})}^{2}$ .

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Step 5.3.1.1

Apply the product rule to $\frac{x}{16}$ .

$x = \frac{x^{2}}{16^{2}}$

Step 5.3.1.2

Raise $16$ to the power of $2$ .

$x = \frac{x^{2}}{256}$

Step 6

Solve for $x$ .

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Step 6.1

Multiply both sides by $256$ .

$x \cdot 256 = \frac{x^{2}}{256} \cdot 256$

Step 6.2

Simplify.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Move $256$ to the left of $x$ .

$256 x = \frac{x^{2}}{256} \cdot 256$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Cancel the common factor of $256$ .

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Step 6.2.2.1.1

Cancel the common factor.

$256 x = \frac{x^{2}}{256} \cdot 256$

Step 6.2.2.1.2

Rewrite the expression.

$256 x = x^{2}$

Step 6.3

Solve for $x$ .

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Step 6.3.1

Subtract $x^{2}$ from both sides of the equation.

$256 x - x^{2} = 0$

Step 6.3.2

Factor the left side of the equation.

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Step 6.3.2.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$256 u - u^{2} = 0$

Step 6.3.2.2

Factor $u$ out of $256 u - u^{2}$ .

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Step 6.3.2.2.1

Factor $u$ out of $256 u$ .

$u \cdot 256 - u^{2} = 0$

Step 6.3.2.2.2

Factor $u$ out of $- u^{2}$ .

$u \cdot 256 + u (- u) = 0$

Step 6.3.2.2.3

Factor $u$ out of $u \cdot 256 + u (- u)$ .

$u (256 - u) = 0$

Step 6.3.2.3

Replace all occurrences of $u$ with $x$ .

$x (256 - x) = 0$

Step 6.3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$256 - x = 0$

Step 6.3.4

Set $x$ equal to $0$ .

$x = 0$

Step 6.3.5

Set $256 - x$ equal to $0$ and solve for $x$ .

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Step 6.3.5.1

Set $256 - x$ equal to $0$ .

$256 - x = 0$

Step 6.3.5.2

Solve $256 - x = 0$ for $x$ .

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Step 6.3.5.2.1

Subtract $256$ from both sides of the equation.

$- x = - 256$

Step 6.3.5.2.2

Divide each term in $- x = - 256$ by $- 1$ and simplify.

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Step 6.3.5.2.2.1

Divide each term in $- x = - 256$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 256}{- 1}$

Step 6.3.5.2.2.2

Simplify the left side.

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Step 6.3.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 256}{- 1}$

Step 6.3.5.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 256}{- 1}$

Step 6.3.5.2.2.3

Simplify the right side.

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Step 6.3.5.2.2.3.1

Divide $- 256$ by $- 1$ .

$x = 256$

Step 6.3.6

The final solution is all the values that make $x (256 - x) = 0$ true.

$x = 0, 256$