Problem 11 - Intermediate Algebra

Question

Evaluate each of the following. For example, \(\sqrt{25}=5\). \(\sqrt{\frac{16}{25}}\)

Step-by-Step Solution

Answer

\(\sqrt{\frac{16}{25}} = \frac{4}{5}\)

Step 1: Understand the Problem

The task is to find the square root of the fraction \( \frac{16}{25} \). This means we need to find a number which, when multiplied by itself, gives \( \frac{16}{25} \).

Step 2: Decompose the Fraction

Consider the fraction \( \frac{16}{25} \). Each part of the fraction should be broken down into its numerator and denominator and addressed separately. Here, 16 and 25 are both perfect squares.

Step 3: Find the Square Root of the Numerator

Identify the square root of 16. The number 4 is the square root of 16 because \( 4 \times 4 = 16 \).

Step 4: Find the Square Root of the Denominator

Identify the square root of 25. The number 5 is the square root of 25 because \( 5 \times 5 = 25 \).

Step 5: Calculate the Square Root of the Fraction

The square root of a fraction \( \sqrt{\frac{a}{b}} \) can be calculated by taking the square root of both the numerator and the denominator separately. Thus, \( \sqrt{\frac{16}{25}} = \frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5} \).

Step 6: Verify the Solution

Verify that the solution \( \frac{4}{5} \) is correct by ensuring that \( \left( \frac{4}{5} \right)^2 = \frac{16}{25} \). Calculate the square: \( \left( \frac{4}{5} \right)^2 = \frac{4 \times 4}{5 \times 5} = \frac{16}{25} \). The calculation confirms the solution is correct.

Key Concepts

Perfect Squares

In mathematics, a perfect square is the product of an integer multiplied by itself. For example, numbers like 4, 9, 16, and 25 are perfect squares because they can be written as \(2 \times 2\), \(3 \times 3\), \(4 \times 4\), and \(5 \times 5\) respectively. Recognizing perfect squares is crucial when dealing with square root calculations, like the one in our exercise with \(\sqrt{\frac{16}{25}}\).
Since both 16 and 25 are perfect squares, finding their square roots becomes straightforward:

• The square root of 16 is 4, because \(4 \times 4 = 16\).
• The square root of 25 is 5, because \(5 \times 5 = 25\).

Recognizing perfect squares quickly allows us to simplify square roots more efficiently.

Fractional Exponents

Fractional exponents are a way to express roots in terms of exponents. They are incredibly useful for simplifying expressions and solving equational problems. For instance, the square root of a number can be represented as that number raised to the power of 1/2. So, for a number \(a\), the square root can be written as \(a^{1/2}\).
In the context of fractions, understanding fractional exponents means realizing that:

• \(\sqrt{\frac{16}{25}} = \left(\frac{16}{25}\right)^{1/2}\)
• This can be separately applied to both the numerator and the denominator due to the properties of exponents: \(\left(16^{1/2}\right) / \left(25^{1/2}\right)\).

Thus, using fractional exponents can often simplify the evaluation of square roots, especially for more complex expressions.

Numerator and Denominator Square Root

When taking the square root of a fraction, you can handle the numerator and denominator separately. This is particularly beneficial when both numbers are perfect squares.
For the fraction \(\frac{16}{25}\), both 16 (numerator) and 25 (denominator) are perfect squares. This method simplifies the problem:

• Calculate the square root of the numerator: \(\sqrt{16} = 4\).
• Calculate the square root of the denominator: \(\sqrt{25} = 5\).

Hence, the square root of the fraction simplifies to \(\frac{4}{5}\). This process of separating the components makes solving square roots of fractions much simpler and avoids any unnecessary complexity.