Simplify square root of y/11

The question is asking to perform a mathematical simplification on a given expression. The expression is the square root of a fraction where 'y' is the numerator and '11' is the denominator. The task is to simplify this square root without actually calculating the value of 'y' since it appears to be a variable and not a specific number. The simplification process may involve rationalizing the denominator or simplifying the expression in a way that makes it easier to understand or further manipulate algebraically.

$\sqrt{\frac{y}{11}}$

Answer

Expert–verified

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

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

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: $\frac{\sqrt{11 y}}{11}$ .

Knowledge Notes:

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

Radical Properties: The square root of a fraction can be expressed as the fraction of the square roots of the numerator and denominator: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ .
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.
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 $(a^{m})^{n} = a^{m n}$ .
- The square of a square root eliminates the radical: $(\sqrt{a})^{2} = a$ .
Product Rule for Radicals: The square root of a product is equal to the product of the square roots: $\sqrt{a b} = \sqrt{a} \cdot \sqrt{b}$ .
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.