Algebra Examples

$\sqrt{8} - \sqrt{32} - \sqrt{18} = 2 \sqrt{a} - 4 \sqrt{a} - 3 \sqrt{a}$

Step 1

Rewrite the equation as $2 \sqrt{a} - 4 \sqrt{a} - 3 \sqrt{a} = \sqrt{8} - \sqrt{32} - \sqrt{18}$ .

$2 \sqrt{a} - 4 \sqrt{a} - 3 \sqrt{a} = \sqrt{8} - \sqrt{32} - \sqrt{18}$

Step 2

Solve for $\sqrt{a}$ .

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Step 2.1

Simplify $2 \sqrt{a} - 4 \sqrt{a} - 3 \sqrt{a}$ .

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Step 2.1.1

Subtract $4 \sqrt{a}$ from $2 \sqrt{a}$ .

$- 2 \sqrt{a} - 3 \sqrt{a} = \sqrt{8} - \sqrt{32} - \sqrt{18}$

Step 2.1.2

Subtract $3 \sqrt{a}$ from $- 2 \sqrt{a}$ .

$- 5 \sqrt{a} = \sqrt{8} - \sqrt{32} - \sqrt{18}$

Step 2.2

Simplify $\sqrt{8} - \sqrt{32} - \sqrt{18}$ .

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.2.1.1.1

Factor $4$ out of $8$ .

$- 5 \sqrt{a} = \sqrt{4 (2)} - \sqrt{32} - \sqrt{18}$

Step 2.2.1.1.2

Rewrite $4$ as $2^{2}$ .

$- 5 \sqrt{a} = \sqrt{2^{2} \cdot 2} - \sqrt{32} - \sqrt{18}$

Step 2.2.1.2

Pull terms out from under the radical.

$- 5 \sqrt{a} = 2 \sqrt{2} - \sqrt{32} - \sqrt{18}$

Step 2.2.1.3

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 2.2.1.3.1

Factor $16$ out of $32$ .

$- 5 \sqrt{a} = 2 \sqrt{2} - \sqrt{16 (2)} - \sqrt{18}$

Step 2.2.1.3.2

Rewrite $16$ as $4^{2}$ .

$- 5 \sqrt{a} = 2 \sqrt{2} - \sqrt{4^{2} \cdot 2} - \sqrt{18}$

Step 2.2.1.4

Pull terms out from under the radical.

$- 5 \sqrt{a} = 2 \sqrt{2} - (4 \sqrt{2}) - \sqrt{18}$

Step 2.2.1.5

Multiply $4$ by $- 1$ .

$- 5 \sqrt{a} = 2 \sqrt{2} - 4 \sqrt{2} - \sqrt{18}$

Step 2.2.1.6

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 2.2.1.6.1

Factor $9$ out of $18$ .

$- 5 \sqrt{a} = 2 \sqrt{2} - 4 \sqrt{2} - \sqrt{9 (2)}$

Step 2.2.1.6.2

Rewrite $9$ as $3^{2}$ .

$- 5 \sqrt{a} = 2 \sqrt{2} - 4 \sqrt{2} - \sqrt{3^{2} \cdot 2}$

Step 2.2.1.7

Pull terms out from under the radical.

$- 5 \sqrt{a} = 2 \sqrt{2} - 4 \sqrt{2} - (3 \sqrt{2})$

Step 2.2.1.8

Multiply $3$ by $- 1$ .

$- 5 \sqrt{a} = 2 \sqrt{2} - 4 \sqrt{2} - 3 \sqrt{2}$

Step 2.2.2

Simplify by adding terms.

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Step 2.2.2.1

Subtract $4 \sqrt{2}$ from $2 \sqrt{2}$ .

$- 5 \sqrt{a} = - 2 \sqrt{2} - 3 \sqrt{2}$

Step 2.2.2.2

Subtract $3 \sqrt{2}$ from $- 2 \sqrt{2}$ .

$- 5 \sqrt{a} = - 5 \sqrt{2}$

Step 2.3

Divide each term in $- 5 \sqrt{a} = - 5 \sqrt{2}$ by $- 5$ and simplify.

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Step 2.3.1

Divide each term in $- 5 \sqrt{a} = - 5 \sqrt{2}$ by $- 5$ .

$\frac{- 5 \sqrt{a}}{- 5} = \frac{- 5 \sqrt{2}}{- 5}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $- 5$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{- 5 \sqrt{a}}{- 5} = \frac{- 5 \sqrt{2}}{- 5}$

Step 2.3.2.1.2

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

$\sqrt{a} = \frac{- 5 \sqrt{2}}{- 5}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Cancel the common factor of $- 5$ .

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Step 2.3.3.1.1

Cancel the common factor.

$\sqrt{a} = \frac{- 5 \sqrt{2}}{- 5}$

Step 2.3.3.1.2

Divide $\sqrt{2}$ by $1$ .

$\sqrt{a} = \sqrt{2}$

Step 3

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

${\sqrt{a}}^{2} = {\sqrt{2}}^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

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

${(a^{\frac{1}{2}})}^{2} = {\sqrt{2}}^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

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

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Step 4.2.1.1.1

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

$a^{\frac{1}{2} \cdot 2} = {\sqrt{2}}^{2}$

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

$a^{\frac{1}{2} \cdot 2} = {\sqrt{2}}^{2}$

Step 4.2.1.1.2.2

Rewrite the expression.

$a^{1} = {\sqrt{2}}^{2}$

Step 4.2.1.2

Simplify.

$a = {\sqrt{2}}^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

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

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Step 4.3.1.1

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

$a = {(2^{\frac{1}{2}})}^{2}$

Step 4.3.1.2

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

$a = 2^{\frac{1}{2} \cdot 2}$

Step 4.3.1.3

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

$a = 2^{\frac{2}{2}}$

Step 4.3.1.4

Cancel the common factor of $2$ .

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Step 4.3.1.4.1

Cancel the common factor.

$a = 2^{\frac{2}{2}}$

Step 4.3.1.4.2

Rewrite the expression.

$a = 2^{1}$

Step 4.3.1.5

Evaluate the exponent.

$a = 2$