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Chapter 8: Problem 18

Multiply. (Assume all variables are non negative.) $$ 4 \sqrt{3} \cdot 2 \sqrt{3} $$

Short Answer

Expert verified
The result is 24.

Step by step solution

01

Multiply the Constants

First, identify the constant numbers outside the square roots, which are 4 and 2. Multiply these constants together: \[ 4 \times 2 = 8 \]
02

Multiply the Radicals

Next, identify the numbers inside the radicals (square roots). These are both 3, so multiply them together inside the square root: \[ \sqrt{3} \times \sqrt{3} = \sqrt{9} \]
03

Simplify the Radical

Simplify the radical \( \sqrt{9} \). Since the square root of 9 is 3, replace \( \sqrt{9} \) with 3: \[ \sqrt{9} = 3 \]
04

Multiply the Results

Now multiply the result of Step 1 by the simplified result from Step 3:\[ 8 \times 3 = 24 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

constants multiplication
Multiplying constants is a fundamental skill in mathematics, especially when dealing with expressions that include square roots. When you have expressions like \(4 \sqrt{3} \cdot 2 \sqrt{3}\), the constants refer to the numerical factors outside of the square root symbols. In our case, these constants are 4 and 2. To multiply them, simply multiply the numbers as you normally would in arithmetic:
  • Step 1: Identify the constants, which are 4 and 2 here.
  • Step 2: Multiply them together: \(4 \times 2 = 8\).
Once multiplied, this result (8 in this case) will become part of your final answer after combining it with other steps in the process. This step simplifies the expression while keeping the constants in check.
radical simplification
Radical simplification is an important step when dealing with expressions under square roots. When you multiply radicals, especially those that are the same, such as \(\sqrt{3}\), you're working on simplifying them. In the step-by-step solution for our example \(4 \sqrt{3} \cdot 2 \sqrt{3}\), you saw:
  • The radicals \(\sqrt{3}\) and \(\sqrt{3}\) were multiplied together: \(\sqrt{3} \times \sqrt{3} = \sqrt{9}\).
  • This feels like multiplying the contents of the square root, leading to another square root.
  • Recognize that \(\sqrt{a} \times \sqrt{a} = a\) which simplifies the problem greatly.
Understanding this rule makes tackling similar problems easier. Applying this rule helps you reduce complex expressions into simpler forms.
square roots
Square roots are special elements in algebra that provide the principal squared number that equals the original number. For example, \(\sqrt{9} = 3\) because \(3^2 = 9\). When you deal with multiplying radicals like \(4 \sqrt{3} \cdot 2 \sqrt{3}\), simplifying the square root at the end becomes crucial:
  • After multiplying, you get \(\sqrt{9}\).
  • The square root of 9 is 3.
Simplifying square roots often leads to whole numbers, simplifying further calculations. Mastering square roots includes recognizing perfect squares and being able to calculate their roots quickly.
algebraic expressions
An algebraic expression is a mathematical statement that can contain numbers, variables, and operations. In the example \(4 \sqrt{3} \cdot 2 \sqrt{3}\), you are dealing with a more complex algebraic expression which involves these elements:
  • There are constants, which are the numbers without square roots.
  • There are radicals (square roots) which need simplification.
The goal with algebraic expressions like this is to combine like terms. This means multiplying constants together and simplifying radicals as much as possible. Once you've simplified each segment, your final expression, \(24\) in this example, will be fully simplified. Understanding these expressions empowers you to handle similar complex mathematical tasks with confidence.

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Most popular questions from this chapter

Simplify. (Assume all radicands containing variable expressions are positive.) \(-8 a b--\sqrt{4}+3 a b--\sqrt{4}-2 a b--\sqrt{4}\)

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