Problem 328 - Elementary Algebra

Question

In the following exercises, simplify. $$ \frac{\sqrt{48}}{\sqrt{75}} $$

Step-by-Step Solution

Answer

\( \frac{\sqrt{48}}{\sqrt{75}} \) simplifies to \( \frac{4}{5} \).

Step 1: Simplify the Radicals

First, we can simplify both the numerator and the denominator independently by factoring the numbers inside the radicals into their prime factors. \ \ \[- \sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4 \sqrt{3} \] \ \ \[- \sqrt{75} = \sqrt{25 \cdot 3} = \sqrt{25} \cdot \sqrt{3} = 5 \sqrt{3} \]

Step 2: Divide the Simplified Radicals

Next, we divide the simplified radicals from Step 1. \ \[- \frac{\sqrt{48}}{\sqrt{75}} = \frac{4 \sqrt{3}}{5 \sqrt{3}} \]

Step 3: Cancel Out Common Factors

The \( \sqrt{3} \) terms in the numerator and the denominator cancel each other out: \ \[- \frac{4 \sqrt{3}}{5 \sqrt{3}} = \frac{4}{5} \]

Key Concepts

Prime Factorization

Let's start by understanding prime factorization. Prime factorization involves breaking down a composite number into its prime factors, which are numbers that can only be divided by 1 and themselves. For example, the prime factorization of 48 is:
- 48 = 2 x 2 x 2 x 2 x 3
These are the prime numbers that multiply together to give 48. Similarly, the prime factorization of 75 is:
- 75 = 5 x 5 x 3
Understanding prime factorization helps in simplifying radicals by making the numbers under the radical sign smaller and more manageable.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. When simplifying a radical expression, we seek to express the number under the radical (the radicand) in its simplest form. For instance, \(\frac{\text{\textbackslash sqrt{48}}}{\text{\sqrt{75}}}\) can be simplified by breaking down each radicand using their prime factors:
- \(\text{\sqrt{48}} = \text{\textbackslash sqrt{16} \textbackslash cdot 3} = \text{\textbackslash sqrt{16}} \textbackslash cdot \text{\textbackslash sqrt{3}} = 4\text{\sqrt{3}}\)
- \(\text{\sqrt{75}} = \text{\textbackslash sqrt{25} \cdot 3} = \text{\sqrt{25}} \textbackslash cdot \text{\textbackslash sqrt{3}} = 5 \text{\sqrt{3}}\)
Now, simplifying these expressions makes it easier to divide and solve them further.

Canceling Common Factors

Canceling common factors is an essential step in simplifying fractions, including those with radicals. After simplifying both the numerator and the denominator to \(\frac{4\text{\sqrt{3}}}{5\text{\sqrt{3}}}\), we notice that \(\text{\sqrt{3}}\) appears in both the numerator and the denominator. These common factors can be canceled out:
- \(\frac{4 \texthat \text{\textbackslash sqrt{3}}}{5 \texthat \text{\textbackslash sqrt{3}}}\) becomes \(\frac{4}{5}\)
By canceling out \(\texthat \text{\textbackslash sqrt{3}}\), the fraction simplifies to \(\frac{4}{5}\), making the radicals easier to manage and the expression much simpler.