What is this expression in simplified form?
√108/√3
A. 6
B. 3
C. 36
D. √105
The Correct Answer and Explanation is:
The expression (\frac{\sqrt{108}}{\sqrt{3}}) can be simplified using the properties of square roots. Specifically, when dividing square roots, we can write it as the square root of the quotient of the two numbers.
So, (\frac{\sqrt{108}}{\sqrt{3}} = \sqrt{\frac{108}{3}}).
Next, we simplify (\frac{108}{3}):
[
108 \div 3 = 36
]
This means the expression simplifies to:
[
\sqrt{\frac{108}{3}} = \sqrt{36}
]
Now, (\sqrt{36}) is a perfect square, as (36) is (6) squared. So,
[
\sqrt{36} = 6
]
Therefore, the correct answer is:
A. 6
Explanation
The simplification process here relies on the properties of square roots, particularly the quotient rule, which states that (\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}), provided that (b) is not zero. Applying this rule to the expression (\frac{\sqrt{108}}{\sqrt{3}}) allowed us to combine the square roots into one by dividing 108 by 3 under a single square root symbol. This transformation simplifies our calculation and eliminates the need to evaluate each square root separately.
After dividing, we obtained (\sqrt{36}). Recognizing that 36 is a perfect square ((6 \times 6 = 36)), we could then evaluate (\sqrt{36}) directly as 6. If 36 had not been a perfect square, we would have either left it in radical form or simplified it to the extent possible. In this problem, the initial square root operation simplified to an integer because 36 is a perfect square.
The answer, therefore, is 6, or option A. This approach illustrates how simplifying complex square root expressions often involves using the rules of radicals, especially the quotient rule, to transform the expression into something easier to evaluate.