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Chapter 11: Problem 9

Perform the indicated operations, expressing answers in simplest form with rationalized denominators. $$\sqrt[3]{4} \sqrt[3]{2}$$

Short Answer

Expert verified
\( \sqrt[3]{4} \sqrt[3]{2} = 2 \).

Step by step solution

01

Identify the Problem

We are given the expression \( \sqrt[3]{4} \times \sqrt[3]{2} \) and need to find the result in simplest form.
02

Use the Properties of Radicals

According to the properties of radicals, \( \sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{ab} \). So here, we can write \( \sqrt[3]{4} \times \sqrt[3]{2} = \sqrt[3]{4 \times 2} \).
03

Simplify the Expression

Now, multiply the numbers inside the cube root: \( 4 \times 2 = 8 \), resulting in \( \sqrt[3]{8} \).
04

Further Simplify Using Cube Root Properties

We need to simplify \( \sqrt[3]{8} \). The cube root of 8 is 2 because \( 2^3 = 8 \). Therefore, \( \sqrt[3]{8} = 2 \).

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Key Concepts

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

Properties of Radicals
Radicals are symbols that indicate the root of a number. The most common types are square roots, cube roots, and higher roots. Each type of radical has properties that help in simplifying expressions. Here are some basic properties:
  • The product property states that for any non-negative numbers \(a\) and \(b\), and any positive integer \(n\), \( \sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{ab} \). This can simplify expressions by combining the terms under a single radical.
  • The quotient property lets us divide the numbers inside the radical: \( \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \).
  • Radicals can also be manipulated similarly to powers. For instance, \( \sqrt[n]{a^m} = a^{\frac{m}{n}} \), and this can transform under-root problems into easier power problems.
Understanding these properties is a key step in working with radicals and making complex expressions more straightforward to simplify.
Rationalized Denominators
A rationalized denominator refers to ensuring the denominator of a fraction is a simple, non-radical number. This practice comes from the need to simplify expressions in a way that they are easier to understand or compare.
  • For instance, if you have \(\frac{1}{\sqrt{2}}\), you can multiply numerator and denominator by \(\sqrt{2}\) to get \(\frac{\sqrt{2}}{2}\).
  • The process involves multiplying by a form of 1, such as \(\frac{\sqrt[n]{b}}{\sqrt[n]{b}}\), effectively ridding the denominator of a radical by making it a rational number.
  • This process is useful in further simplifying solutions to ensure they are expressible using integers in the denominator.
Rationalizing denominators can help make results of radical operations manageable and easy to use in further computations.
Cube Root Properties
The cube root is a specific type of radical, represented by \(\sqrt[3]{a}\), which asks "what number, when multiplied by itself twice, equals \(a\)?" Understanding some properties of cube roots can make solving such problems easier.
  • The cube root of a product: For numbers \(a\) and \(b\), \( \sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b} \). This property allows for the simplification seen in the given problem.
  • Cubes and cube roots walk hand-in-hand through expressions like \(2^3 = 8\) meaning \(\sqrt[3]{8} = 2\). Recognizing cube numbers such as 8, 27, 64, helps streamline simplification.
  • Unlike square roots, cube roots can be negative because multiplying a negative number by itself three times results in a negative number, e.g., \((-2)^3 = -8\) so \(\sqrt[3]{-8} = -2\).
By mastering these cube root properties, solving problems involving cube roots becomes a manageable task with straightforward steps to reach the simplest form.

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Most popular questions from this chapter

Express each of the given expressions in simplest form with only positive exponents. $$\left(R_{1}^{-1}+R_{2}^{-1}\right)^{-1}$$

Solve the given problems. If \(x<0,\) is it ever true that \(x^{-2}

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$$\text {Solve the given problems.}$$ One leg of a right triangle is \(2 \sqrt{7}\) and the hypotenuse is \(6 .\) What is the area of the triangle?

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