Problem 57 - College Algebra with Corequisite Support

Question

For the following exercises, simplify each expression. \(\sqrt{\frac{81 m}{361 m^{2}}}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{9}{19\sqrt{m}} \).

Step 1: Simplify the Fraction Under the Square Root

First, look at the fraction \( \frac{81m}{361m^2} \). Separate it into two parts: \( \frac{81}{361} \) and \( \frac{m}{m^2} \).

Step 2: Simplify \( \frac{81}{361} \)

Identify that \( 81 \) and \( 361 \) are perfect squares. \( 81 \) is \( 9^2 \) and \( 361 \) is \( 19^2 \). Thus, \( \frac{81}{361} = \frac{9^2}{19^2} \).

Step 3: Simplify \( \frac{m}{m^2} \)

Recognize that dividing powers with the same base allows the subtraction of exponents. So, \( \frac{m}{m^2} = m^{1-2} = m^{-1} \).

Step 4: Recombine and Simplify the Expression Under the Square Root

Combine the results from steps 2 and 3 into the square root: \( \sqrt{\frac{9^2}{19^2} \cdot m^{-1}} \). This becomes \( \sqrt{\frac{9^2}{19^2} \cdot \frac{1}{m}} \).

Step 5: Take the Square Root Separately

Taking the square root of a fraction allows treating the numerator and denominator separately: \( \sqrt{\frac{9^2}{19^2} \cdot \frac{1}{m}} = \frac{9}{19} \cdot \frac{1}{\sqrt{m}} \).

Step 6: Simplify and Present the Final Simplified Expression

The final simplified expression from taking the square roots separately is \( \frac{9}{19\sqrt{m}} \). You can also rationalize the denominator to obtain an alternative form.

Key Concepts

Understanding Square Roots

A square root is a number that, when multiplied by itself, gives the original number. It essentially "undoes" the squaring of a number, bringing us back to the initial value. The square root symbol, \(\sqrt{}\), is used to represent it. For example, \(\sqrt{81}\) is \(9\) because \(9 \times 9 = 81\).

In simplifying expressions with square roots, especially when dealing with fractions, we can separate the square root of the numerator and the denominator. This allows us to simplify expressions more effectively. For instance, \(\sqrt{\frac{81m}{361m^2}}\) can be simplified by first looking at \(\sqrt{81}\) and \(\sqrt{361}\) individually. Recognizing perfect squares helps to simplify these further. The square root of a perfect square is an integer, which makes calculations and simplifications more manageable.

When expressions involve variables, such as \(m\), understanding and applying the laws of exponents will also play a role in simplification. Remember to always work toward ensuring the expression inside the square root is in its simplest form before proceeding.

Rationalization of Denominators

Rationalization is a process applied to remove square roots (or other irrational numbers) from the denominator of a fraction. When a square root is in the denominator, it can make further calculations more complex, so rationalization simplifies this.

To rationalize a denominator like in the expression \(\frac{9}{19\sqrt{m}}\), we multiply both the numerator and the denominator by the square root that is present in the denominator. This keeps the fraction equivalent to the original, while getting rid of the radical in the denominator. For example:

• The given fraction is \(\frac{9}{19\sqrt{m}}\).
• Multiply both numerator and denominator by \(\sqrt{m}\), giving \(\frac{9\sqrt{m}}{19m}\).

Now, the expression has a rational denominator, which makes it simpler to work with in subsequent calculations. Be mindful of maintaining equivalence; the operations affect both numerator and denominator similarly to preserve the value of the fraction.

Simplification of Fractions

Fraction simplification involves reducing fractions to their simplest form, where the numerator and denominator share no common factors aside from 1.

When simplifying the fraction \(\frac{81m}{361m^2}\), we first look for common factors in numbers and use exponent rules for variables:

• Identify \(81\) as \(9^2\) and \(361\) as \(19^2\). Divide both the numerator and denominator by their greatest common factor, sharpening the simplification process.
• For the variables, apply the subtraction rule: \(\frac{m}{m^2} = m^{1-2} = m^{-1}\). This leaves a single variable \(m\) in the form of a negative exponent.

By breaking down each component, you simplify either separately before recombining, or directly within the continued simplifying. Remember that simplifying fractions not only makes them easier to read but also helps avoid potential mistakes in more complex mathematical operations.