Simplify square root of 2/( square root of 3)

Solution:

Step 1:

Rationalize the denominator of $\frac{\sqrt{2}}{\sqrt{3}}$ by multiplying by $\frac{\sqrt{3}}{\sqrt{3}}$ to get $\sqrt{\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}}$.

Step 2:

Simplify the expression within the square root.

Step 2.1:

Multiply numerators and denominators to get $\sqrt{\frac{2 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}}}$.

Step 2.2:

Recognize that $\sqrt{3}$ raised to the first power is $\sqrt{3}$, thus $\sqrt{\frac{2 \cdot \sqrt{3}}{(\sqrt{3})^1 \cdot \sqrt{3}}}$.

Step 2.3:

Repeat the recognition for the other $\sqrt{3}$ to obtain $\sqrt{\frac{2 \cdot \sqrt{3}}{(\sqrt{3})^1 \cdot (\sqrt{3})^1}}$.

Step 2.4:

Apply the exponent rule $a^m \cdot a^n = a^{m+n}$ to combine the exponents, resulting in $\sqrt{\frac{2 \cdot \sqrt{3}}{(\sqrt{3})^{1+1}}}$.

Step 2.5:

Add the exponents to get $\sqrt{\frac{2 \cdot \sqrt{3}}{(\sqrt{3})^2}}$.

Step 2.6:

Simplify the denominator.

Step 2.6.1:

Express $\sqrt{3}$ as $3^{\frac{1}{2}}$ and rewrite as $\sqrt{\frac{2 \cdot \sqrt{3}}{((3^{\frac{1}{2}}))^2}}$.

Step 2.6.2:

Apply the exponent multiplication rule to get $\sqrt{\frac{2 \cdot \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}}$.

Step 2.6.3:

Multiply the exponents to simplify $\sqrt{\frac{2 \cdot \sqrt{3}}{3^{\frac{2}{2}}}}$.

Step 2.6.4:

Cancel out the common factors to obtain $\sqrt{\frac{2 \cdot \sqrt{3}}{3^1}}$.

Step 2.6.5:

Evaluate the simplified exponent to get $\sqrt{\frac{2 \cdot \sqrt{3}}{3}}$.

Step 3:

Express the square root of the fraction as a fraction of square roots: $\frac{\sqrt{2 \cdot \sqrt{3}}}{\sqrt{3}}$.

Step 4:

Rationalize the denominator of $\frac{\sqrt{2 \cdot \sqrt{3}}}{\sqrt{3}}$ by multiplying by $\frac{\sqrt{3}}{\sqrt{3}}$.

Step 5:

Simplify the expression within the denominator.

Step 5.1:

Multiply numerators and denominators to get $\frac{\sqrt{2 \cdot \sqrt{3} \cdot \sqrt{3}}}{\sqrt{3} \cdot \sqrt{3}}$.

Step 5.2:

Recognize that $\sqrt{3}$ raised to the first power is $\sqrt{3}$, thus $\frac{\sqrt{2 \cdot \sqrt{3} \cdot \sqrt{3}}}{(\sqrt{3})^1 \cdot \sqrt{3}}$.

Step 5.3:

Repeat the recognition for the other $\sqrt{3}$ to obtain $\frac{\sqrt{2 \cdot \sqrt{3} \cdot \sqrt{3}}}{(\sqrt{3})^1 \cdot (\sqrt{3})^1}$.

Step 5.4:

Apply the exponent rule $a^m \cdot a^n = a^{m+n}$ to combine the exponents, resulting in $\frac{\sqrt{2 \cdot \sqrt{3} \cdot \sqrt{3}}}{(\sqrt{3})^{1+1}}$.

Step 5.5:

Add the exponents to get $\frac{\sqrt{2 \cdot \sqrt{3} \cdot \sqrt{3}}}{(\sqrt{3})^2}$.

Step 5.6:

Simplify the denominator.

Step 5.6.1:

Express $\sqrt{3}$ as $3^{\frac{1}{2}}$ and rewrite as $\frac{\sqrt{2 \cdot \sqrt{3} \cdot \sqrt{3}}}{((3^{\frac{1}{2}}))^2}$.

Step 5.6.2:

Apply the exponent multiplication rule to get $\frac{\sqrt{2 \cdot \sqrt{3} \cdot \sqrt{3}}}{3^{\frac{1}{2} \cdot 2}}$.

Step 5.6.3:

Multiply the exponents to simplify $\frac{\sqrt{2 \cdot \sqrt{3} \cdot \sqrt{3}}}{3^{\frac{2}{2}}}$.

Step 5.6.4:

Cancel out the common factors to obtain $\frac{\sqrt{2 \cdot \sqrt{3} \cdot \sqrt{3}}}{3^1}$.

Step 5.6.5:

Evaluate the simplified exponent to get $\frac{\sqrt{2 \cdot \sqrt{3} \cdot \sqrt{3}}}{3}$.

Step 6:

Simplify the numerator.

Step 6.1:

Combine using the product rule for radicals to get $\frac{\sqrt{2 \cdot \sqrt{3} \cdot 3}}{3}$.

Step 6.2:

Multiply inside the radical to obtain $\frac{\sqrt{6 \cdot \sqrt{3}}}{3}$.

Step 7:

The final result can be presented in various forms.

Exact Form: $\frac{\sqrt{6 \cdot \sqrt{3}}}{3}$ Decimal Form: $1.07456993 \ldots$

Knowledge Notes:

To simplify the square root of a fraction, we can use the property that the square root of a quotient is the quotient of the square roots. However, when the denominator is irrational (like $\sqrt{3}$), we often rationalize the denominator to make it a rational number. This is done by multiplying the fraction by a form of one that will eliminate the square root in the denominator.

The steps taken in this solution involve rationalizing the denominator, simplifying the expression within the square root, and using exponent rules to further simplify the expression. The power rule for exponents states that $a^m \cdot a^n = a^{m+n}$, and the rule for raising a power to a power is $(a^m)^n = a^{m \cdot n}$. These rules are used to combine and simplify exponents within the problem.

Additionally, the product rule for radicals, which states that $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$, is used to combine square roots into a single square root when possible. This helps in simplifying the expression further.

The final result is given in both exact form, which includes the radical, and decimal form, which is an approximation of the exact value.