Algebra Examples

$\frac{10 - \sqrt{18}}{\sqrt{2}} = a + b \sqrt{2}$

Step 1

Rewrite the equation as $a + b \sqrt{2} = \frac{10 - \sqrt{18}}{\sqrt{2}}$ .

$a + b \sqrt{2} = \frac{10 - \sqrt{18}}{\sqrt{2}}$

Step 2

Simplify $\frac{10 - \sqrt{18}}{\sqrt{2}}$ .

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Step 2.1

Simplify the numerator.

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Step 2.1.1

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

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Step 2.1.1.1

Factor $9$ out of $18$ .

$a + b \sqrt{2} = \frac{10 - \sqrt{9 (2)}}{\sqrt{2}}$

Step 2.1.1.2

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

$a + b \sqrt{2} = \frac{10 - \sqrt{3^{2} \cdot 2}}{\sqrt{2}}$

Step 2.1.2

Pull terms out from under the radical.

$a + b \sqrt{2} = \frac{10 - (3 \sqrt{2})}{\sqrt{2}}$

Step 2.1.3

Multiply $3$ by $- 1$ .

$a + b \sqrt{2} = \frac{10 - 3 \sqrt{2}}{\sqrt{2}}$

Step 2.2

Multiply $\frac{10 - 3 \sqrt{2}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$a + b \sqrt{2} = \frac{10 - 3 \sqrt{2}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 2.3

Combine and simplify the denominator.

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Step 2.3.1

Multiply $\frac{10 - 3 \sqrt{2}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$a + b \sqrt{2} = \frac{(10 - 3 \sqrt{2}) \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 2.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$a + b \sqrt{2} = \frac{(10 - 3 \sqrt{2}) \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 2.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$a + b \sqrt{2} = \frac{(10 - 3 \sqrt{2}) \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 2.3.4

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

$a + b \sqrt{2} = \frac{(10 - 3 \sqrt{2}) \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 2.3.5

Add $1$ and $1$ .

$a + b \sqrt{2} = \frac{(10 - 3 \sqrt{2}) \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 2.3.6

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

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Step 2.3.6.1

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

$a + b \sqrt{2} = \frac{(10 - 3 \sqrt{2}) \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 2.3.6.2

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

$a + b \sqrt{2} = \frac{(10 - 3 \sqrt{2}) \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 2.3.6.3

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

$a + b \sqrt{2} = \frac{(10 - 3 \sqrt{2}) \sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.3.6.4

Cancel the common factor of $2$ .

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Step 2.3.6.4.1

Cancel the common factor.

$a + b \sqrt{2} = \frac{(10 - 3 \sqrt{2}) \sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.3.6.4.2

Rewrite the expression.

$a + b \sqrt{2} = \frac{(10 - 3 \sqrt{2}) \sqrt{2}}{2^{1}}$

Step 2.3.6.5

Evaluate the exponent.

$a + b \sqrt{2} = \frac{(10 - 3 \sqrt{2}) \sqrt{2}}{2}$

Step 2.4

Apply the distributive property.

$a + b \sqrt{2} = \frac{10 \sqrt{2} - 3 \sqrt{2} \sqrt{2}}{2}$

Step 2.5

Multiply $- 3 \sqrt{2} \sqrt{2}$ .

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Step 2.5.1

Raise $\sqrt{2}$ to the power of $1$ .

$a + b \sqrt{2} = \frac{10 \sqrt{2} - 3 ({\sqrt{2}}^{1} \sqrt{2})}{2}$

Step 2.5.2

Raise $\sqrt{2}$ to the power of $1$ .

$a + b \sqrt{2} = \frac{10 \sqrt{2} - 3 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{2}$

Step 2.5.3

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

$a + b \sqrt{2} = \frac{10 \sqrt{2} - 3 {\sqrt{2}}^{1 + 1}}{2}$

Step 2.5.4

Add $1$ and $1$ .

$a + b \sqrt{2} = \frac{10 \sqrt{2} - 3 {\sqrt{2}}^{2}}{2}$

Step 2.6

Simplify each term.

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Step 2.6.1

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

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Step 2.6.1.1

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

$a + b \sqrt{2} = \frac{10 \sqrt{2} - 3 {(2^{\frac{1}{2}})}^{2}}{2}$

Step 2.6.1.2

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

$a + b \sqrt{2} = \frac{10 \sqrt{2} - 3 \cdot 2^{\frac{1}{2} \cdot 2}}{2}$

Step 2.6.1.3

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

$a + b \sqrt{2} = \frac{10 \sqrt{2} - 3 \cdot 2^{\frac{2}{2}}}{2}$

Step 2.6.1.4

Cancel the common factor of $2$ .

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Step 2.6.1.4.1

Cancel the common factor.

$a + b \sqrt{2} = \frac{10 \sqrt{2} - 3 \cdot 2^{\frac{2}{2}}}{2}$

Step 2.6.1.4.2

Rewrite the expression.

$a + b \sqrt{2} = \frac{10 \sqrt{2} - 3 \cdot 2^{1}}{2}$

Step 2.6.1.5

Evaluate the exponent.

$a + b \sqrt{2} = \frac{10 \sqrt{2} - 3 \cdot 2}{2}$

Step 2.6.2

Multiply $- 3$ by $2$ .

$a + b \sqrt{2} = \frac{10 \sqrt{2} - 6}{2}$

Step 2.7

Cancel the common factor of $10 \sqrt{2} - 6$ and $2$ .

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Step 2.7.1

Factor $2$ out of $10 \sqrt{2}$ .

$a + b \sqrt{2} = \frac{2 (5 \sqrt{2}) - 6}{2}$

Step 2.7.2

Factor $2$ out of $- 6$ .

$a + b \sqrt{2} = \frac{2 (5 \sqrt{2}) + 2 \cdot - 3}{2}$

Step 2.7.3

Factor $2$ out of $2 (5 \sqrt{2}) + 2 (- 3)$ .

$a + b \sqrt{2} = \frac{2 (5 \sqrt{2} - 3)}{2}$

Step 2.7.4

Cancel the common factors.

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Step 2.7.4.1

Factor $2$ out of $2$ .

$a + b \sqrt{2} = \frac{2 (5 \sqrt{2} - 3)}{2 (1)}$

Step 2.7.4.2

Cancel the common factor.

$a + b \sqrt{2} = \frac{2 (5 \sqrt{2} - 3)}{2 \cdot 1}$

Step 2.7.4.3

Rewrite the expression.

$a + b \sqrt{2} = \frac{5 \sqrt{2} - 3}{1}$

Step 2.7.4.4

Divide $5 \sqrt{2} - 3$ by $1$ .

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

Step 3

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

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