Algebra Examples

$\sqrt{3} q + 2 = \sqrt{5}$

Step 1

Subtract $2$ from both sides of the equation.

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

Step 2

Divide each term in $\sqrt{3} q = \sqrt{5} - 2$ by $\sqrt{3}$ and simplify.

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Step 2.1

Divide each term in $\sqrt{3} q = \sqrt{5} - 2$ by $\sqrt{3}$ .

$\frac{\sqrt{3} q}{\sqrt{3}} = \frac{\sqrt{5}}{\sqrt{3}} + \frac{- 2}{\sqrt{3}}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $\sqrt{3}$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{\sqrt{3} q}{\sqrt{3}} = \frac{\sqrt{5}}{\sqrt{3}} + \frac{- 2}{\sqrt{3}}$

Step 2.2.1.2

Divide $q$ by $1$ .

$q = \frac{\sqrt{5}}{\sqrt{3}} + \frac{- 2}{\sqrt{3}}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $\frac{\sqrt{5}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$q = \frac{\sqrt{5}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} + \frac{- 2}{\sqrt{3}}$

Step 2.3.1.2

Combine and simplify the denominator.

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Step 2.3.1.2.1

Multiply $\frac{\sqrt{5}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$q = \frac{\sqrt{5} \sqrt{3}}{\sqrt{3} \sqrt{3}} + \frac{- 2}{\sqrt{3}}$

Step 2.3.1.2.2

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

$q = \frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} + \frac{- 2}{\sqrt{3}}$

Step 2.3.1.2.3

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

$q = \frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} + \frac{- 2}{\sqrt{3}}$

Step 2.3.1.2.4

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

$q = \frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1 + 1}} + \frac{- 2}{\sqrt{3}}$

Step 2.3.1.2.5

Add $1$ and $1$ .

$q = \frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{2}} + \frac{- 2}{\sqrt{3}}$

Step 2.3.1.2.6

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

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Step 2.3.1.2.6.1

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

$q = \frac{\sqrt{5} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} + \frac{- 2}{\sqrt{3}}$

Step 2.3.1.2.6.2

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

$q = \frac{\sqrt{5} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} + \frac{- 2}{\sqrt{3}}$

Step 2.3.1.2.6.3

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

$q = \frac{\sqrt{5} \sqrt{3}}{3^{\frac{2}{2}}} + \frac{- 2}{\sqrt{3}}$

Step 2.3.1.2.6.4

Cancel the common factor of $2$ .

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Step 2.3.1.2.6.4.1

Cancel the common factor.

$q = \frac{\sqrt{5} \sqrt{3}}{3^{\frac{2}{2}}} + \frac{- 2}{\sqrt{3}}$

Step 2.3.1.2.6.4.2

Rewrite the expression.

$q = \frac{\sqrt{5} \sqrt{3}}{3^{1}} + \frac{- 2}{\sqrt{3}}$

Step 2.3.1.2.6.5

Evaluate the exponent.

$q = \frac{\sqrt{5} \sqrt{3}}{3} + \frac{- 2}{\sqrt{3}}$

Step 2.3.1.3

Simplify the numerator.

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Step 2.3.1.3.1

Combine using the product rule for radicals.

$q = \frac{\sqrt{5 \cdot 3}}{3} + \frac{- 2}{\sqrt{3}}$

Step 2.3.1.3.2

Multiply $5$ by $3$ .

$q = \frac{\sqrt{15}}{3} + \frac{- 2}{\sqrt{3}}$

Step 2.3.1.4

Move the negative in front of the fraction.

$q = \frac{\sqrt{15}}{3} - \frac{2}{\sqrt{3}}$

Step 2.3.1.5

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

$q = \frac{\sqrt{15}}{3} - (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 2.3.1.6

Combine and simplify the denominator.

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Step 2.3.1.6.1

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

$q = \frac{\sqrt{15}}{3} - \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 2.3.1.6.2

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

$q = \frac{\sqrt{15}}{3} - \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 2.3.1.6.3

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

$q = \frac{\sqrt{15}}{3} - \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 2.3.1.6.4

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

$q = \frac{\sqrt{15}}{3} - \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 2.3.1.6.5

Add $1$ and $1$ .

$q = \frac{\sqrt{15}}{3} - \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 2.3.1.6.6

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

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Step 2.3.1.6.6.1

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

$q = \frac{\sqrt{15}}{3} - \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 2.3.1.6.6.2

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

$q = \frac{\sqrt{15}}{3} - \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 2.3.1.6.6.3

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

$q = \frac{\sqrt{15}}{3} - \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.3.1.6.6.4

Cancel the common factor of $2$ .

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Step 2.3.1.6.6.4.1

Cancel the common factor.

$q = \frac{\sqrt{15}}{3} - \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.3.1.6.6.4.2

Rewrite the expression.

$q = \frac{\sqrt{15}}{3} - \frac{2 \sqrt{3}}{3^{1}}$

Step 2.3.1.6.6.5

Evaluate the exponent.

$q = \frac{\sqrt{15}}{3} - \frac{2 \sqrt{3}}{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$q = \frac{\sqrt{15}}{3} - \frac{2 \sqrt{3}}{3}$

Decimal Form:

$q = 0.13629391 \dots$