Problem 24 - Algebra and Trigonometry

Question

Simplify the expression. \(\sqrt{75}+\sqrt{48}\)

Step-by-Step Solution

Answer

The expression simplifies to \(9\sqrt{3}\).

Step 1: Factor the Radicands

First, we need to break down the numbers inside the square roots into their prime factors. For \(75\), the prime factorization is \(75 = 3 imes 5^2\). For \(48\), the prime factorization is \(48 = 2^4 imes 3\).

Step 2: Simplify Each Radical Expression

Use the prime factorization to rewrite each square root. For \(\sqrt{75}\), we have \(\sqrt{3 \times 5^2} = 5\sqrt{3}\) because \(5^2\) comes out of the square root. For \(\sqrt{48}\), we have \(\sqrt{2^4 \times 3} = \sqrt{(2^2)^2 \times 3} = 4\sqrt{3}\) because \(2^2\) comes out of the square root.

Step 3: Combine Like Terms

Both simplified expressions contain \(\sqrt{3}\), so they are like terms and can be combined. Add \(5\sqrt{3} + 4\sqrt{3}\) to get \((5 + 4)\sqrt{3} = 9\sqrt{3}\).

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks, specifically into its prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. This is an essential skill when simplifying radical expressions because it helps identify the numbers that can "come out" of the square root.

• To factor a number like 75, start with the smallest prime, which is 2, and see if it divides evenly. It doesn’t, so move to the next prime, which is 3.
• Since 3 divides 75, you have 75 divided by 3 equals 25. But 25 can further be broken down into 5 times 5.
• This gives us the factorization: 75 = 3 × 5².

Repeat this process with 48:

• 48 can be divided by 2 (the smallest prime) to get 24, which is 2 times 12, and then 12 can be divided further into 2 times 6, finally 6 into 2 times 3.
• So, 48 = 2⁴ × 3.

Understanding prime factorization simplifies other steps considerably, particularly when simplifying square roots as explained in the next section.

Simplifying Square Roots

Simplifying square roots involves rewriting a square root in its simplest form. This often includes taking pairs of identical factors out from under the radical sign, which we uncover using prime factorization. Let's see how this applies to our expression.Starting with \(\sqrt{75}\):

• The prime factorization we found was \(75 = 3 \times 5^2\).
• Inside the square root, any identical pairs (like 5² in this case) can "come out" as a single factor (5).
• This gives us \(\sqrt{75} = 5\sqrt{3}\).

Now for \(\sqrt{48}\):

• The factorization \(48 = 2^4 \times 3\) allows for pairs of 2, written as \((2^2)^2\).
• Therefore, one pair of 2's comes out, which simplifies \(\sqrt{48}\) to \(4\sqrt{3}\).

Using these steps helps to simplify radicals efficiently by reducing them to terms that are easier to handle.

Combining Like Terms

In the context of radical expressions, combining like terms means adding or subtracting those terms that have the same radicand and the same index. For example, in our expression, both terms have the radical part \(\sqrt{3}\), enabling us to combine them.Here is how it works:

• From the previous steps, we simplified \(\sqrt{75}\) to \(5\sqrt{3}\) and \(\sqrt{48}\) to \(4\sqrt{3}\).
• These are like terms because they both include \(\sqrt{3}\). They can be easily added by combining the coefficients (5 and 4) associated with the \(\sqrt{3}\).
• Performing this addition gives us \((5 + 4)\sqrt{3} = 9\sqrt{3}\).

This process highlights why simplifying and combining like terms is such a powerful tool in mathematics, providing clarity and making further calculations easier.