To rationalize the denominator of $\frac{-\sqrt{7}}{4 \sqrt{5}}$, what value should the numerator and denominator be multiplied by?
a) $-\sqrt{7}$
b) $\sqrt{5}$
c) $\sqrt{7}$
d) $4-\sqrt{5}$
Answer (1)
Identify the radical in the denominator: 5 .
Multiply both the numerator and the denominator by 5 to eliminate the radical.
Simplify the expression: 4 5 − 7 ⋅ 5 5 = 20 − 35 .
The value to multiply by is 5 .
Explanation
Understanding the Problem We are given the expression 4 5 − 7 and asked to rationalize the denominator. Rationalizing the denominator means we want to eliminate any radical expressions from the denominator. In this case, we have 4 5 in the denominator, so we need to get rid of the 5 .
Rationalizing the Denominator To eliminate the 5 from the denominator, we can multiply both the numerator and the denominator by 5 . This will give us: 4 5 − 7 ⋅ 5 5 = 4 5 ⋅ 5 − 7 ⋅ 5 = 4 ⋅ 5 − 7 ⋅ 5 = 20 − 35
Value to Multiply By So, we multiplied both the numerator and the denominator by 5 to rationalize the denominator.
Final Answer The value that the numerator and denominator should be multiplied by is 5 . Therefore, the answer is b) 5 .
Examples
Rationalizing the denominator is a technique used to simplify expressions and make them easier to work with. For example, in physics, you might encounter an expression with a square root in the denominator when calculating the electric field due to a charge distribution. Rationalizing the denominator can help you simplify the expression and make it easier to compare with experimental data. Suppose you have an expression like 2 1 in your calculations. To rationalize the denominator, you multiply both the numerator and the denominator by 2 to get 2 2 , which is often easier to work with.
