Algebra Examples

$\sqrt{7} \div \sqrt{3} - \sqrt{2}$

Step 1

Simplify each term.

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Step 1.1

Rewrite the division as a fraction.

$\frac{\sqrt{7}}{\sqrt{3}} - \sqrt{2}$

Step 1.2

Multiply $\frac{\sqrt{7}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\sqrt{7}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} - \sqrt{2}$

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

Multiply $\frac{\sqrt{7}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\sqrt{7} \sqrt{3}}{\sqrt{3} \sqrt{3}} - \sqrt{2}$

Step 1.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{7} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} - \sqrt{2}$

Step 1.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{7} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} - \sqrt{2}$

Step 1.3.4

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

$\frac{\sqrt{7} \sqrt{3}}{{\sqrt{3}}^{1 + 1}} - \sqrt{2}$

Step 1.3.5

Add $1$ and $1$ .

$\frac{\sqrt{7} \sqrt{3}}{{\sqrt{3}}^{2}} - \sqrt{2}$

Step 1.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\sqrt{7} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} - \sqrt{2}$

Step 1.3.6.2

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

$\frac{\sqrt{7} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} - \sqrt{2}$

Step 1.3.6.3

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

$\frac{\sqrt{7} \sqrt{3}}{3^{\frac{2}{2}}} - \sqrt{2}$

Step 1.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.6.4.1

Cancel the common factor.

$\frac{\sqrt{7} \sqrt{3}}{3^{\frac{2}{2}}} - \sqrt{2}$

Step 1.3.6.4.2

Rewrite the expression.

$\frac{\sqrt{7} \sqrt{3}}{3^{1}} - \sqrt{2}$

Step 1.3.6.5

Evaluate the exponent.

$\frac{\sqrt{7} \sqrt{3}}{3} - \sqrt{2}$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Combine using the product rule for radicals.

$\frac{\sqrt{7 \cdot 3}}{3} - \sqrt{2}$

Step 1.4.2

Multiply $7$ by $3$ .

$\frac{\sqrt{21}}{3} - \sqrt{2}$

Step 2

To write $- \sqrt{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\sqrt{21}}{3} - \sqrt{2} \cdot \frac{3}{3}$

Step 3

Combine $- \sqrt{2}$ and $\frac{3}{3}$ .

$\frac{\sqrt{21}}{3} + \frac{- \sqrt{2} \cdot 3}{3}$

Step 4

Simplify the expression.

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Step 4.1

Combine the numerators over the common denominator.

$\frac{\sqrt{21} - \sqrt{2} \cdot 3}{3}$

Step 4.2

Multiply $3$ by $- 1$ .

$\frac{\sqrt{21} - 3 \sqrt{2}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{21} - 3 \sqrt{2}}{3}$

Decimal Form:

$0.11331166 \dots$