Simplify square root of 120/( square root of 3)

Solution:

Step 1:

Rationalize the denominator by multiplying $\frac{120}{\sqrt{3}}$ with $\frac{\sqrt{3}}{\sqrt{3}}$ to get $\sqrt{\frac{120}{\sqrt{3}}\cdot \frac{\sqrt{3}}{\sqrt{3}}}$.

Step 2:

Simplify the expression inside the square root.

Step 2.1:

Multiply the numerator and denominator by $\sqrt{3}$ to obtain $\sqrt{\frac{120\sqrt{3}}{3}}$.

Step 2.2:

Recognize that $\sqrt{3}\cdot \sqrt{3}=3$ and rewrite the expression as $\sqrt{\frac{120\sqrt{3}}{3}}$.

Step 2.3:

Simplify the denominator to get $\sqrt{\frac{120\sqrt{3}}{3}}$.

Step 3:

Reduce the fraction inside the square root by dividing both the numerator and the denominator by $3$.

Step 3.1:

Extract $3$ from $120\sqrt{3}$ to get $\sqrt{\frac{3\cdot 40\sqrt{3}}{3}}$.

Step 3.2:

Cancel out the common factor of $3$ to simplify to $\sqrt{40\sqrt{3}}$.

Step 4:

Express $40$ as $4\cdot 10$ and recognize that $4$ is a perfect square.

Step 4.1:

Factor out $4$ from $40$ to get $\sqrt{4\cdot 10\sqrt{3}}$.

Step 4.2:

Rewrite $4$ as $2^2$ to obtain $\sqrt{2^2\cdot 10\sqrt{3}}$.

Step 4.3:

Group the terms under the square root as $\sqrt{2^2\cdot (10\sqrt{3})}$.

Step 5:

Extract the square root of the perfect square, $2^2$, from under the radical to get $2\sqrt{10\sqrt{3}}$.

Step 6:

Present the final result in both exact and decimal forms.

Exact Form: $2\sqrt{10\sqrt{3}}$ Decimal Form: $8.32358290\dots$

Knowledge Notes:

To simplify a square root expression, especially when it involves a fraction, we use the following knowledge points:

1. Rationalizing the Denominator: To eliminate square roots from the denominator, we multiply the fraction by a form of one that contains the square root in both the numerator and the denominator.

2. Simplifying Square Roots: We simplify square roots by identifying and extracting perfect squares and reducing fractions under the radical when possible.

3. Power Rules: We use the power rules for exponents, such as $a^m\cdot a^n=a^{m+n}$ and $(a^m{)}^n=a^{mn}$, to simplify expressions involving powers.

4. Square Root Properties: We apply properties like $\sqrt{a\cdot b}=\sqrt{a}\cdot \sqrt{b}$ and $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$ to break down and simplify square roots.

5. Perfect Squares: Recognizing and extracting perfect squares from under the square root helps in simplifying the expression. For instance, $\sqrt{4}=2$ because $4$ is a perfect square.

By applying these principles, we can systematically simplify square root expressions and rationalize denominators to arrive at a simplified form, both exact and approximate (decimal).