Problem 36 - College Algebra

Question

In Exercises \(27-38,\) add or subtract terms whenever possible. $$4 \sqrt{12}-2 \sqrt{75}$$

Step-by-Step Solution

Answer

The result of \(4 \sqrt{12}-2 \sqrt{75}\) is \(-2\sqrt{3}\)

Step 1: Simplify First Term

First, simplify \(4 \sqrt{12}\). The prime factorization of 12 is \(2^2*3\). Therefore, we can simplify the square root by taking the square number (\(2^2\)) out of the root sign to get \(4*2*\sqrt{3}=8\sqrt{3}\)

Step 2: Simplify Second Term

Second, simplify \(2\sqrt{75}\). The number 75 can be factorized into \(3*5^2\). Similarly, we take out the square \(5^2\) from the square root to get \(2*5*\sqrt{3}=10\sqrt{3}\)

Step 3: Perform Subtraction

Now subtract the simplified second term from the first term to get \(8\sqrt{3}-10\sqrt{3}\). The result is \(-2\sqrt{3}\).

Key Concepts

Simplifying Radicals

Simplifying radicals is an essential skill when dealing with square roots and algebraic expressions. The goal is to simplify the square root by finding numbers that can be squared and taken out of the root. Here's how you can do it:

• Prime Factorization: Break down the number inside the square root into its prime factors.
• Identify Squares: Look for pairs of prime factors that create perfect squares.
• Extract the Perfect Squares: Take these squares outside of the square root as whole numbers.

For instance, to simplify \(\sqrt{12}\), find its prime factorization, which is \(2^2 \times 3\). You can take \(2^2\) out of the root, simplifying it to \(2\sqrt{3}\). This process helps reduce complex expressions into simpler forms, making calculations more manageable.

Subtraction of Terms

The subtraction of terms in algebra involves combining like terms, which are terms that have the same variable part. In the context of radicals, like terms have the same radicand, or the number inside the square root. Here's how you subtract them:

• Simplify Each Term: Before subtracting, ensure each radical is simplified. This involves reducing the radicals to their simplest form.
• Combine Like Terms: Subtract coefficients of radicals with the same radicand.
• Avoid Different Radicands: You can't subtract radicals with different radicands directly.

For example, if you have \(8\sqrt{3} - 10\sqrt{3}\), you subtract the coefficients \(8\) and \(10\) because the radicand is \(3\) in both terms. This subtraction gives you \(-2\sqrt{3}\). It's crucial to first simplify each radical for accurate subtraction.

Prime Factorization

Prime factorization is a technique used to simplify numbers by breaking them down into prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. Here's how to use prime factorization:

• Identify the Number: Start with the number you want to factorize.
• Find Primes: Determine smaller prime numbers that can divide the number.
• Repeat Division: Continue dividing the quotient by prime numbers until you reach 1.

Consider the number 75, factorized into \(3 \times 5^2\). This representation helps identify pairs of factors, such as \(5^2\), which can be extracted from radicals easily. By expressing numbers as products of primes, you simplify calculations involving radicals and make operations such as simplification or subtraction more straightforward.