Simplify square root of 120/( square root of 3)
The question asks to simplify the mathematical expression that consists of a square root in the numerator and a square root in the denominator. Specifically, it requires simplifying the square root of 120 divided by the square root of 3. The simplification process would likely involve rationalizing the denominator and simplifying any square roots by factoring out perfect squares, if possible. The goal is to express the initial complex fraction as a simpler or more standard form.
$\sqrt{\frac{120}{\sqrt{3}}}$
Answer
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: $8.32358290 \ldots$
Knowledge Notes:
To simplify a square root expression, especially when it involves a fraction, we use the following knowledge points:
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.
Simplifying Square Roots: We simplify square roots by identifying and extracting perfect squares and reducing fractions under the radical when possible.
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.
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.
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).
