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Chapter 9: Problem 15

Change each radical to simplest radical form. $$\frac{\sqrt[3]{375}}{\sqrt[3]{216}}$$

Short Answer

Expert verified
\( \frac{5 \sqrt[3]{3}}{6} \)

Step by step solution

01

Express each radicand as a product of prime factors

To simplify the radicals, we first express 375 and 216 as products of their prime factors. \[ 375 = 3 imes 5^3 \] and \[ 216 = 2^3 imes 3^3 \].
02

Simplify the cube roots

Next, take the cube root of each radicand by extracting the cube powers: \[ \sqrt[3]{375} = \sqrt[3]{3 \times 5^3} = 5 \cdot \sqrt[3]{3} \] and \[ \sqrt[3]{216} = \sqrt[3]{2^3 \times 3^3} = 2 \times 3 = 6 \].
03

Form the fraction using simplified radicals

Substitute the simplified radicals back into the fraction: \[ \frac{\sqrt[3]{375}}{\sqrt[3]{216}} = \frac{5 \cdot \sqrt[3]{3}}{6} \].
04

Simplify the fraction

The expression can be simplified to \[ \frac{5}{6} \cdot \sqrt[3]{3} \]. Thus, the simplest radical form is \[ \frac{5 \sqrt[3]{3}}{6} \].

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Key Concepts

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

Cube Roots
Understanding cube roots is essential when simplifying radical expressions like the one in the exercise above. A cube root asks you which number multiplied by itself three times gives you the original number. For example, the cube root of 8 is 2 because
  • 2 脳 2 脳 2 = 8
Cube roots are represented by the radical symbol with a small 3, written as \( \sqrt[3]{a} \). This is in contrast to square roots, which do not have this additional number.
To solve \( \frac{\sqrt[3]{375}}{\sqrt[3]{216}} \), we need to find the cube roots of the numbers in the numerator and the denominator, starting by breaking them into their smallest factors.
Prime Factorization
Prime factorization helps us break down numbers into their core components, or prime numbers, which are numbers only divisible by 1 and themselves. This method is often used to simplify radical expressions.
For example, let's look at the number 375:
  • The first step is to divide it by the smallest prime number, which is 3, resulting in 125 (since 3 脳 125 = 375).
  • Next, recognize that 125 can be further broken down into \( 5 \times 5 \times 5 \) or 5 cubed, making \( 375 = 3 \times 5^3 \).
Similarly, for 216, following its factorization:
  • Divide by 2 to get 108, then continue splitting it by 2 until resolving into \( 2^3 \times 3^3 \).
This conversion is key as it allows us to identify and simplify cube roots efficiently.
Simplest Radical Form
The simplest radical form requires breaking down an expression fully and presenting it in the most reduced form possible. For cube roots, this means extracting factors that are raised to the power of three.
In our exercise, we factorized and simplified using the cube root rule:
  • The cube root of \( 375 = 3 \times 5^3 \) results in \( 5 \times \sqrt[3]{3} \).
  • For \( 216 = 2^3 \times 3^3 \), it simplifies directly to 6, being \( 2 \times 3 \).
Combine these to revise the fraction: \( \frac{5 \cdot \sqrt[3]{3}}{6} \).
The expression is simplified further to \( \frac{5}{6} \cdot \sqrt[3]{3} \), which is considered the simplest radical form because it's concise and entirely expressed with the smallest possible numbers and roots.

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Most popular questions from this chapter

Use a calculator to find a whole number approximation for each expression. $$\sqrt{8050}$$

Use a calculator to evaluate each radical. (Objective 1) $$\sqrt{2304}$$

Find a rational approximation, to the nearest tenth, for each radical expression. (Objective 2) $$4 \sqrt{3}-2 \sqrt{2}$$

Use a calculator to evaluate each radical. (Objective 1) $$\sqrt[3]{1728}$$

How could you find a whole number approximation for \(\sqrt{1450}\) if you did not have a calculator available?
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