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Chapter 8: Problem 22

Multiply. (Assume all variables are non negative.) $$ (24 \sqrt{3})(34 \sqrt{3}) $$

Short Answer

Expert verified
The product is 2448.

Step by step solution

01

Multiply the Numerical Coefficients

First, we multiply the numerical coefficients of the given expressions. Multiply 24 by 34.\[ 24 \times 34 = 816 \]
02

Multiply the Square Roots

Next, multiply the square roots in the given expressions. Since \( \sqrt{3} \times \sqrt{3} = 3 \), we have: \[ \sqrt{3} \times \sqrt{3} = 3 \]
03

Combine the Results

Now, combine the results from Step 1 and Step 2. Multiply the product of the numerical coefficients by the result from the square roots.\[ 816 \times 3 = 2448 \]
04

Final Answer

The final result of multiplying \((24 \sqrt{3})\) and \((34 \sqrt{3})\) is \(2448\).

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Key Concepts

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

Numerical Coefficients
Numerical coefficients are the numbers you multiply that are outside the square root in a radical expression. In the expression \(24 \sqrt{3} \), the number 24 is the numerical coefficient. Similarly, in \(34 \sqrt{3}\), 34 is the numerical coefficient. To solve the original problem of multiplying radical expressions, our first step is dealing with these numerical coefficients.

Here鈥檚 how you can do it:
  • Pick out the numerical parts of each expression: 24 and 34.
  • Multiply them together: \( 24 \times 34 = 816 \).

This result, 816, represents the combined numerical coefficient that you'll use in the final answer. Remember, when you multiply, begin with these numerical parts to simplify the process. Once that is set, we can focus on the square roots.
Square Roots
Square roots present the second key aspect of multiplying radical expressions. The square root applies to numbers or expressions inside it. Here, the problem involves \( \sqrt{3} \times \sqrt{3} \).

The rules for multiplying square roots are a bit different than regular numbers.
  • When multiplying two identical square roots, the result is simply the number inside the square root.
  • Therefore, \( \sqrt{3} \times \sqrt{3} = 3 \).

It's essential to remember this process when dealing with similar problems. By knowing and applying this property, you can significantly simplify and solve radical multiplication effortlessly.
Combining Results
After we have dealt with both the numerical coefficients and the square roots, the last step is combining the results. This step helps in getting the final answer to our problem.

Let's piece together what we found:
  • From the numerical coefficients: 816.
  • From the square roots: 3.

Now, multiply these two results together: \( 816 \times 3 = 2448 \). This gives you the final answer.

Combining results effectively wraps up all previous calculations into a single, understandable number. It consolidates your work into a neat conclusion, showcasing the power of accurately multiplying both numerical coefficients and square roots.

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Most popular questions from this chapter

Simplify. $$8 x 3 y--\sqrt{3}-2 x \cdot 8 y-\sqrt{3}+27 x 3 y---\sqrt{3}+x \cdot y \sqrt{3}$$

Given the function, calculate the following. $$ g(x)=x \sqrt{3}, \text { find } g(-1), g(0), \text { and } g(1) $$

The y-intercepts for any graph will have the form \((0, y),\) where \(y\) is a real number. Therefore, to find \(y\) -intercepts, set \(x=0\) and solve for \(y .\) Find the \(y\) -intercepts for the following. $$ y=x-1----\sqrt{3}+2 $$

Rewrite the following as a radical expression with coefficient \(1 .\) $$ 52 x--\sqrt{3} $$

Simplify. (Assume all radicands containing variable expressions are positive.) \(418 n 2 m----\sqrt{-} 2 n 8 m---\sqrt{+} n 2 m---\sqrt{1}\)
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