Problem 71 - Algebra 1: Concepts and Skills

Question

Write the radical expression in simplest form. $$ \sqrt{\frac{21}{35}} $$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt{\frac{21}{35}}\) is \(\frac{\sqrt{15}}{5}\)

Step 1: Simplify the Fraction

The fraction \(\frac{21}{35}\) can be simplified by finding a common factor between 21 and 35. In this case, the common factor is 7. Dividing both numerator and denominator by 7 gives us the simplified fraction \(\frac{3}{5}\)

Step 2: Apply the Square Root

Next, apply the square root to both the numerator and the denominator, which can be done as they are now separate terms under the square root. So, \(\sqrt{\frac{3}{5}} = \frac{\sqrt{3}}{\sqrt{5}}\)

Step 3: Rationalize the Denominator

To simplify further, one needs to rationalize the denominator, i.e., eliminate the square root in the denominator. One can do this by multiplying both the numerator and the denominator by \(\sqrt{5}\) which results in \(\frac{\sqrt{3} \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} = \frac{\sqrt{15}}{5}\)

Key Concepts

Understanding Radical Expressions

A radical expression is a mathematical phrase that includes a square root, cube root, or any higher roots. Grasping its structure is crucial for simplification. A radical can contain numbers, variables, or both inside the root symbol, and simplifying them generally involves two steps: combining like terms under the root and reducing the expression into its simplest form.

This simplification process may sometimes include factoring out perfect squares from under the root or applying special rules like the product and quotient rules for radicals. The product rule allows us to multiply the components under the radicals, and similarly, the quotient rule lets us divide them, each resulting in a single radical. This makes it easier to further simplify the expression.

Rationalizing the Denominator

Rationalizing the denominator ensures that radicals are not left in the bottom part of a fraction, which is considered unconventional and mathematically inelegant. The purpose of rationalization is to move any radical from the denominator to the numerator to improve clarity and sometimes to enable further simplification.

One common method to rationalize a square root in the denominator is to multiply the fraction by a form of 1 that contains the same radical in the numerator and denominator. For example, to rationalize \(\frac{1}{\sqrt{a}}\), you can multiply both top and bottom by \(\sqrt{a}\) to end up with \(\frac{\sqrt{a}}{a}\), which has no radical in the denominator.

Simplifying Fractions Within Radicals

When faced with a radical containing a fraction, the goal is to simplify the fraction first before applying the root. This means reducing the fraction to its lowest terms by finding common factors in the numerator and denominator and dividing by them.

Simplifying fractions is a fundamental skill in math, as it often makes the numbers much easier to work with. With simpler numbers, operations such as taking roots become more straightforward. In our example \(\sqrt{\frac{21}{35}}\), we observe that 7 is a common factor. By simplifying the fraction to \(\frac{3}{5}\), we then consider the square root of the numerator and denominator separately before finally rationalizing the denominator.