Which expressions are equivalent to the given expression? $\frac{\sqrt{27}}{\sqrt{3}}$
I. $\sqrt{9}$
II. 3
III. $\frac{3}{\sqrt{3}}$
A. I and II only
B. II only
C. III only
D. I only
Answer (1)
Simplify the given expression: 3 27 = 3 .
Evaluate expression I: 9 = 3 .
Expression II is 3.
Evaluate expression III: 3 3 = 3 .
Expressions I and II are equivalent to the given expression, so the answer is A .
Explanation
Problem Analysis We are asked to determine which of the given expressions are equivalent to 3 27 . Let's analyze each expression step by step.
Simplifying the Given Expression First, let's simplify the given expression: 3 27 = 3 27 = 9 = 3
Evaluating Expression I Now, let's evaluate expression I: 9 = 3 So, expression I is equivalent to the given expression.
Evaluating Expression II Expression II is simply 3, which is also equivalent to the given expression.
Evaluating Expression III Now, let's evaluate expression III: 3 3 To rationalize the denominator, we multiply both the numerator and the denominator by 3 : 3 3 × 3 3 = 3 3 3 = 3 Since 3 = 3 , expression III is not equivalent to the given expression.
Conclusion Therefore, expressions I and II are equivalent to the given expression.
Examples
Understanding equivalent expressions is crucial in various fields, such as engineering and physics, where simplifying complex formulas is essential for efficient calculations. For example, when calculating the impedance of an electrical circuit, you might encounter complex expressions involving square roots and fractions. Simplifying these expressions using the properties of radicals and rationalization can make the calculations more manageable and lead to a clearer understanding of the circuit's behavior. This skill is also useful in everyday situations, such as comparing prices or calculating discounts, where simplifying expressions can help you make informed decisions.
