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Chapter 5: Problem 35

Change each radical to simplest radical form. \(\frac{3}{2} \sqrt{24}\)

Short Answer

Expert verified
The simplest form of \( \frac{3}{2} \sqrt{24} \) is \( 3 \sqrt{6} \).

Step by step solution

01

Identify the Radical Component

The expression is \( \frac{3}{2} \sqrt{24} \). We need to simplify \( \sqrt{24} \).
02

Prime Factorization of the Radicand

Break down the number 24 into its prime factors: 24 = 2 \times 2 \times 2 \times 3 or \( 24 = 2^3 \times 3 \).
03

Simplify the Square Root

The square root of a product that involves powers can be simplified. Rewrite \( \sqrt{24} \) as \( \sqrt{2^3 \times 3} \). Since \( 2^3 = 2^2 \times 2 \), we can simplify \( \sqrt{2^2 \times 2 \times 3} \) to \( 2 \sqrt{2 \times 3} \). Thus, \( \sqrt{24} = 2 \sqrt{6} \).
04

Substitute and Simplify the Expression

Substitute the simplified radical back into the original expression: The expression becomes \( \frac{3}{2} \times 2 \sqrt{6} \).Since the coefficient 2 cancels with the denominator, the expression simplifies to: \( 3 \sqrt{6} \).

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Key Concepts

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

Prime Factorization
Prime factorization is a technique used to break down a number into the prime numbers that multiply together to create the original number. It's like finding the building blocks of a number. For the radicand 24, we use this method to simplify the radical. To find the prime factors, start dividing by the smallest prime number 2 and continue until you can't divide evenly anymore. For 24:
  • 24 divided by 2 is 12
  • 12 divided by 2 is 6
  • 6 divided by 2 is 3
  • 3 is a prime number, so we stop here
Putting it all together: 24 = 2 脳 2 脳 2 脳 3, or written with exponents, 24 = 2^3 脳 3. Understanding the prime factorization helps in simplifying radicals through square root operations.
Square Root Simplification
Once we have the prime factorization, the next step is to simplify the square root. Simplifying means reducing the expression so it's written in the simplest radical form. For a square root like \( \sqrt{24} \), look at the prime factors: \( 24 = 2^3 \times 3 \).
Break down this expression based on the square root properties:
  • Think of \( 2^3 \) as \( 2^2 \times 2 \)
  • Take the square root of \( 2^2 \), which is 2
  • So \( \sqrt{2^2 \times 2 \times 3} = 2 \sqrt{2 \times 3} \)
Hence, \( \sqrt{24} = 2 \sqrt{6} \). This method lets you take out perfect squares from under the radical sign.
Radical Expressions
A radical expression involves roots, such as square roots. Simplifying these expressions requires understanding the properties of radicals and coefficients. Starting with our example: \( \frac{3}{2} \sqrt{24} \), it can be broken into simpler parts:
Initially, simplify the radical \( \sqrt{24} \) to \( 2 \sqrt{6} \). Replace the simplified form back into our expression:
  • Substitute as \( \frac{3}{2} \times 2 \sqrt{6} \)
  • The 2 cancels out with the denominator 2
  • Leaving us with \( 3 \sqrt{6} \)
This final expression, \( 3 \sqrt{6} \), is simpler and easier to interpret. In handling radical expressions, always check for factors that can be simplified by taking square roots, and remember to cancel out any common terms afterward.

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Most popular questions from this chapter

For Problems \(19-32\), write each of the following in ordinary decimal notation. For example, \((3.18)(10)^{2}=318\). \((4.3)(10)^{-1}\)

Use your calculator to estimate each of the following to the nearest one- thousandth. (a) \(7^{\frac{4}{3}}\) (b) \(10^{\frac{4}{5}}\) (c) \(12^{\frac{3}{5}}\) (d) \(19^{\frac{2}{5}}\) (e) \(7^{\frac{3}{4}}\) (f) \(10^{\frac{5}{4}}\)

Write each of the following in scientific notation. For example \(27800=(2.78)(10)^{4}\). \(376.4\)

Use scientific notation and the properties of exponents to help you perform the following operations. \((90,000)^{\frac{3}{2}}\)

Why do we need scientific notation even when using calculators and computers?
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