Problem 57 - College Algebra

Question

Simplify each expression. $$\sqrt{\frac{81 m}{361 m^{2}}}$$

Step-by-Step Solution

Answer

\(\frac{9}{19m}\)

Step 1: Simplify Inside the Square Root

First, simplify the fraction under the square root: \[\frac{81m}{361m^2}\]The fraction consists of the numbers 81 and 361, and the variables \(m\) and \(m^2\).

Step 2: Simplify the Numbers

Next, simplify the numerical part of the fraction. The number 81 is a perfect square since \(9^2 = 81\), and 361 is also a perfect square since \(19^2 = 361\). Therefore:\[\sqrt{\frac{81}{361}} = \frac{9}{19}\]

Step 3: Simplify the Variables

Now, simplify the variable part of the expression. Given \(m\) in the numerator and \(m^2\) in the denominator, you simplify it as follows:\[\frac{m}{m^2} = \frac{1}{m}\]

Step 4: Combine and Simplify the Entire Expression

Combine the simplified numerical and variable parts:\[\sqrt{\frac{81m}{361m^2}} = \sqrt{\frac{9}{19} \cdot \frac{1}{m}} = \frac{9}{19m}\]

Step 5: Simplify the Square Root

Since the square root of the product simplifies to the product of the square roots, the final expression becomes:\[\frac{9}{19m}\]

Key Concepts

Square Root Simplification

Simplifying square roots can often make expressions more manageable and easier to work with. When you see a square root sign \( \sqrt{\cdot} \), you are looking to find what number, when squared, gives the result inside. For instance, the square root of 81 is 9 because \( 9^2 = 81 \). This principle helps us break down more complex expressions under a square root.

In the given problem, \( \sqrt{\frac{81m}{361m^2}} \), the fractions inside the square root can be split into their individual components — namely, their numerical and variable parts. When you have a fraction under a square root, like \( \sqrt{\frac{a}{b}} \), you can transform it into \( \frac{\sqrt{a}}{\sqrt{b}} \). This simplification is very useful, as you can work on the numerator and denominator separately.

Fraction Simplification

Fraction simplification involves reducing fractions to their simplest form. This ensures that calculations are straightforward and errors are minimized. To simplify a fraction, you divide both the numerator and the denominator by their greatest common divisor (GCD).

Returning to our problem's numerical part, which is \( \frac{81}{361} \), these numbers are perfect squares (81 and 361 are the squares of 9 and 19, respectively). Therefore, the simplified form becomes \( \frac{9}{19} \). Simplifying fractions like these is crucial as it lays the foundation for solving the entire problem.

Variable Simplification

When it comes to simplifying variables, the key is to follow the laws of exponents. Having terms like \( \frac{m}{m^2} \) means you are allowed to subtract the exponents in the division. This results in \( m^{1-2} = m^{-1} \).

However, to make expressions clear and simple, it is best to write them without negative exponents. So, \( m^{-1} \) can be rewritten as \( \frac{1}{m} \). By reducing the expression neatly, students find it easier to grasp without getting tangled in complex operations. You see, transforming these variables prepares them for being combined seamlessly with the rest of the expression.

Numerical Part of Expressions

Understanding the numerical part of expressions goes beyond just stripping components down to basics. It involves recognizing perfect squares and other mathematical properties, making it easier to spot possibilities for simplification.

In our problem, spotting that 81 and 361 are perfect squares allows us to cleanly break down the expression into a simpler form. This recognition helps in separating and organizing different parts of more complex expressions, which can then be recombined in a simplified version, such as the final \( \frac{9}{19m} \). Breaking down numbers perfectly paves the way for easier interpretation and faster computation.