Simplify. $\newline$ $\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{2}}$

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Algebra 1

Multiply radical expressions

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Q. Simplify.

\newline

\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{2}}

Find Common Denominator: We are given the expression $(1)/(\sqrt{3}) + (1)/(\sqrt{2})$ . To add these fractions, we need a common denominator. The common denominator for $\sqrt{3}$ and $\sqrt{2}$ is $\sqrt{3}\cdot\sqrt{2} = \sqrt{6}$ .
Rewrite Fractions: Now we will rewrite each fraction with the common denominator of $\sqrt{6}$ . To do this, we multiply the numerator and denominator of each fraction by the necessary form of $1$ to get the common denominator without changing the value of the fractions. $\frac{1}{\sqrt{3}} \times \frac{\sqrt{2}}{\sqrt{2}} + \frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{2}}{\sqrt{6}} + \frac{\sqrt{3}}{\sqrt{6}}$
Add Numerators: Now that we have a common denominator, we can add the numerators. $\frac{\sqrt{2} + \sqrt{3}}{\sqrt{6}}$
Rationalize Denominator: To simplify the expression further, we can rationalize the denominator by multiplying the numerator and the denominator by $\sqrt{6}$ . $\frac{\sqrt{2} + \sqrt{3}}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{\sqrt{2}\sqrt{6} + \sqrt{3}\sqrt{6}}{\sqrt{6}\sqrt{6}}$
Simplify Numerator: Now we simplify the numerator and the denominator. $\frac{\sqrt{12} + \sqrt{18}}{6}$
Factor Square Numbers: We can simplify $\sqrt{12}$ and $\sqrt{18}$ further by factoring out the square numbers. $\newline$ $\sqrt{12} = \sqrt{4\times3} = \sqrt{4}\times\sqrt{3} = 2\times\sqrt{3}$ $\newline$ $\sqrt{18} = \sqrt{9\times2} = \sqrt{9}\times\sqrt{2} = 3\times\sqrt{2}$
Substitute Simplified Forms: Now we substitute the simplified forms of $\sqrt{12}$ and $\sqrt{18}$ back into the expression. $\newline$ $(2\cdot\sqrt{3} + 3\cdot\sqrt{2})/(6)$
Check Further Simplification: We can now see if there is any further simplification possible. However, since $2$ and $3$ do not have a common factor with $6$ , and $\sqrt{3}$ and $\sqrt{2}$ cannot be simplified further, this is the simplest form of the expression.