Problem 5 - Elementary and Intermediate Algebra

Question

Fill in the blanks by selecting from the following words (which may be used more than once): radicand(s), indices, conjugate(s), base(s) denominator(s), numerator(s). To rationalize the ______ of \(\frac{\sqrt{c}-\sqrt{a}}{5},\) we multiply by a form of \(1,\) using the _____ of \(\sqrt{c}-\sqrt{a},\) or \(\sqrt{c}+\sqrt{a}\) to write 1

Step-by-Step Solution

Answer

To rationalize the denominator, multiply by the conjugate, which is \(\root c + \root a\).

Step 1 - Identify the component to be rationalized

Determine the part of the fraction that needs to be rationalized. Here, it's the denominator of \(\frac{\root c \times \root a}{5}\).

Step 2 - Recall the technique for rationalization

Rationalization involves removing the radicals from the denominator. This is done by multiplying by the conjugate of the given expression.

Step 3 - Identify the conjugate

For \(\root c - \root a\), the conjugate is \(\root c + \root a\). Multiplying the numerator and the denominator by this conjugate will help rationalize the expression.

Step 4 - Write the expression for multiplication

Multiply both the numerator and the denominator by the conjugate. The expression becomes \(\frac{\root c - \root a}{5} \times \frac{\root c + \root a}{\root c + \root a}\)

Key Concepts

rationalizing the denominator

Rationalizing the denominator is a method used to eliminate radicals from the bottom of a fraction.
When a fraction has a square root or other radicals in the denominator, it can make calculations more difficult.
To make the fraction easier to work with, we 'rationalize' it.
This involves multiplying both the numerator and the denominator by a suitable number (or expression) to get rid of the radical in the denominator.
For example, in the fraction \(\frac{\root c - \root a}{5}\), the denominator is 5, which does not need rationalization, but if the denominator was a radical like \(\root c - \root a\), we would need to rationalize it by multiplying the whole fraction by its conjugate.

radicals

Radicals are symbols that represent the root of a number, such as square roots or cube roots.
The most common radical is the square root, denoted by \(\root \).\br> In algebra, radicals can appear in both the numerator and the denominator of fractions.
When working with fractions that involve radicals, we may need to rationalize the expression to make calculations simpler.
For instance, in the fraction \(\frac{\root c - \root a}{5}\), \(\root c\) and \(\root a\) are radicals.
It's often necessary to simplify these expressions to make the math easier to handle.

conjugates

A conjugate is a pair of binomials where one has the same terms, but the sign in between them is different.
For example, the conjugate of \(\root c - \root a\) is \(\root c + \root a\).
Conjugates are particularly useful in rationalizing denominators that contain radicals.
Multiplying a fraction by the conjugate of the denominator helps eliminate the radical from the denominator.
In our example, multiplying \(\frac{\root c - \root a}{5}\) by the conjugate \(\root c + \root a\) can help simplify the fraction.

fractions

Fractions consist of a numerator and a denominator.
The numerator is the top part, and the denominator is the bottom.
Sometimes we'll encounter fractions with radicals in either the numerator or the denominator.
In such cases, we may need to rationalize the fraction to make it easier to work with.
For example, in the fraction \(\frac{\root c - \root a}{5}\), \(\root c - \root a\) is the numerator, and 5 is the denominator.
Multiplying the entire fraction by the conjugate of \(\root c - \root a\), which is \(\root c + \root a\), helps remove the radicals and simplify the expression.