Simplify square root of y/11

Solution:

Step 1:

Express $\sqrt{\frac{y}{11}}$ as $\frac{\sqrt{y}}{\sqrt{11}}$.

Step 2:

Multiply the expression by $\frac{\sqrt{11}}{\sqrt{11}}$ to rationalize the denominator: $\frac{\sqrt{y}}{\sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}}$.

Step 3:

Simplify the denominator by following these sub-steps:

Step 3.1:

Multiply the numerators and denominators: $\frac{\sqrt{y} \cdot \sqrt{11}}{\sqrt{11} \cdot \sqrt{11}}$.

Step 3.2:

Recognize that $\sqrt{11}$ raised to the power of $1$ is still $\sqrt{11}$: $\frac{\sqrt{y} \cdot \sqrt{11}}{(\sqrt{11})^1 \cdot \sqrt{11}}$.

Step 3.3:

Repeat the recognition of the power of $1$: $\frac{\sqrt{y} \cdot \sqrt{11}}{(\sqrt{11})^1 \cdot (\sqrt{11})^1}$.

Step 3.4:

Apply the exponent rule $a^m \cdot a^n = a^{m+n}$: $\frac{\sqrt{y} \cdot \sqrt{11}}{(\sqrt{11})^{1+1}}$.

Step 3.5:

Add the exponents $1 + 1$: $\frac{\sqrt{y} \cdot \sqrt{11}}{(\sqrt{11})^2}$.

Step 3.6:

Transform $(\sqrt{11})^2$ back into $11$ by following these sub-steps:

Step 3.6.1:

Rewrite $\sqrt{11}$ as $(11)^{\frac{1}{2}}$: $\frac{\sqrt{y} \cdot \sqrt{11}}{((11)^{\frac{1}{2}})^2}$.

Step 3.6.2:

Use the power rule $(a^m)^n = a^{mn}$: $\frac{\sqrt{y} \cdot \sqrt{11}}{(11)^{\frac{1}{2} \cdot 2}}$.

Step 3.6.3:

Multiply $\frac{1}{2}$ by $2$: $\frac{\sqrt{y} \cdot \sqrt{11}}{(11)^{\frac{2}{2}}}$.

Step 3.6.4:

Simplify the fraction $\frac{2}{2}$ to $1$.

Step 3.6.4.1:

Cancel out the common factors: $\frac{\sqrt{y} \cdot \sqrt{11}}{(11)^{\frac{\cancel{2}}{\cancel{2}}}}$.

Step 3.6.4.2:

Rewrite the expression as $\frac{\sqrt{y} \cdot \sqrt{11}}{(11)^1}$.

Step 3.6.5:

Evaluate the exponent to get $\frac{\sqrt{y} \cdot \sqrt{11}}{11}$.

Step 4:

Combine the radicals using the product rule: $\frac{\sqrt{y \cdot 11}}{11}$.

Step 5:

Rearrange the factors within the radical: $\frac{\sqrt{11y}}{11}$.

Knowledge Notes:

To simplify the square root of a fraction, you can apply the following knowledge points:

1. Radical Properties: The square root of a fraction can be expressed as the fraction of the square roots of the numerator and denominator: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$.

2. Rationalizing the Denominator: To eliminate the radical from the denominator, you can multiply the fraction by a form of 1 that contains the radical, effectively rationalizing the denominator.

3. Exponent Rules:

   • The power rule for exponents states that $a^m \cdot a^n = a^{m+n}$.

   • The rule for raising a power to a power is $(a^m)^n = a^{mn}$.

   • The square of a square root eliminates the radical: $(\sqrt{a})^2 = a$.

4. Product Rule for Radicals: The square root of a product is equal to the product of the square roots: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$.

5. Simplifying Expressions: When simplifying expressions, it's important to combine like terms and reduce fractions when possible.

By applying these principles, you can simplify the square root of a fraction and rationalize the denominator to obtain an equivalent expression that is easier to work with.