Problem 260 - Elementary Algebra

Question

In the following exercises, simplify. (a) \((4 \sqrt{11})(-3 \sqrt{11})\) (b) \((5 \sqrt{3})^{2}\)

Step-by-Step Solution

Answer

(a) -132, (b) 75

Step 1: Simplify Exercise Part (a)

First, multiply the coefficients: 4 and -3. The result is -12. Next, since both terms have \( \sqrt{11} \), multiply the square roots: \( \sqrt{11} \times \sqrt{11} = \sqrt{121} = 11 \). Finally, combine the results: -12 \times 11 = -132. So, \( (4 \sqrt{11})(-3 \sqrt{11}) = -132 \).

Step 2: Simplify Exercise Part (b)

First, square the coefficient: \( 5^2 = 25 \). Next, square the square root term: \( \sqrt{3}^2 = 3 \). Multiply these results together: 25 \times 3 = 75. So, \( (5 \sqrt{3})^2 = 75 \).

Key Concepts

Multiplying Radicals

Understanding how to multiply radicals is essential to simplify expressions involving square roots.
Let's break it down with a few easy steps.
When multiplying radicals, follow these steps:

• Multiply the coefficients (numbers in front of the radicals).
• Multiply the radicands (numbers inside the square roots).
• Simplify the resulting radical if possible.

For example, in \((4 \sqrt{11})(-3 \sqrt{11})\), we first multiply the coefficients, 4 and -3, to get -12.
Then, multiply the radicands: \sqrt{11} \times \sqrt{11} = \sqrt{121} = 11.
Finally, multiply the results: -12 \times 11 = -132.
So, \((4 \sqrt{11})(-3 \sqrt{11})\) simplifies to -132.

Squaring Radicals

Squaring radicals is similar to multiplying them, but a bit simpler.
When you square a radical expression, you multiply it by itself.
Follow these steps:

• Square the coefficient.
• Square the radicand.

For instance, with \((5 \sqrt{3})^2\), first, square the coefficient: 5^2 = 25.
Then, square the radical: \sqrt{3}^2 = 3.
Finally, multiply these two results: 25 \times 3 = 75.
Thus, \((5 \sqrt{3})^2\) simplifies to 75.

Arithmetic with Coefficients

Combining arithmetic with coefficients and radicals may seem tricky.
But by breaking it down into smaller steps, it becomes manageable.
Here's a clear approach to follow:

• Treat the coefficients and radicals separately during multiplication or division.
• Apply standard arithmetic rules to the coefficients.
• Follow the rules for multiplying or simplifying radicals on the radicands.

In our example of \((4 \sqrt{11})(-3 \sqrt{11})\), we start by focusing on the coefficients: 4 and -3.
The calculation is simple: 4 \times -3 = -12.
Then, work on the radicals: \sqrt{11} \times \sqrt{11} = \sqrt{121} = 11.
Combining these parts gives: -12 \times 11 = -132.
The same applies when squaring expressions such as in \((5 \sqrt{3})^2\).
Handle the coefficient: 5^2 = 25.
Then the radical term: \sqrt{3}^2 = 3.
Combine: 25 \times 3 = 75.
Mastering how to handle coefficients and radicals separately will make these types of problems much easier.