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Chapter 1: Problem 5

Express each in terms of the simplest possible radical. $$\sqrt[3]{4} \times \sqrt[3]{16}$$

Short Answer

Expert verified
The simplest form is 4.

Step by step solution

01

Write the given expression in terms of powers

First, express each cube root as a power. The cube root of a number is the same as raising that number to the 1/3 power. Therefore, \( \sqrt[3]{4} = 4^{1/3} \) and \( \sqrt[3]{16} = 16^{1/3} \).
02

Combine the exponents using properties of exponents

When multiplying numbers with the same base, you add their exponents. However, since the numbers have different bases here, multiply the radicands first and then apply the power. So, \( 4^{1/3} \times 16^{1/3} = (4 \times 16)^{1/3} \).
03

Multiply the radicands

Multiply the numbers inside the cube roots: \( 4 \times 16 = 64 \). So, the expression becomes \( 64^{1/3} \).
04

Simplify the result

Find the cube root of \(64\). Since \(64 = 4^3\), the cube root of \(64\) is \(4\). Therefore, \( 64^{1/3} = 4 \).

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Key Concepts

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

Understanding Cube Roots
The cube root of a number is an operation that asks what number, when multiplied by itself three times, gives you the original number. Think of the cube root as the opposite of cubing a number. For example, if you know that 2 cubed equals 8 (or \(2^3 = 8\)), then the cube root of 8 brings you back to 2. Writing this as an equation, \(\sqrt[3]{8} = 2\). Let's break down how to express this using exponents:
  • A cube root can be turned into an exponent by using a fraction: \(a^{1/3}\)
  • So, \(\sqrt[3]{4} = 4^{1/3}\) and \(\sqrt[3]{16} = 16^{1/3}\) in our exercise.
This makes handling cube roots easier when used together with properties of exponents.
Exploring Properties of Exponents
Properties of exponents help make calculations involving powers simpler. They are vital when multiplying or dividing numbers that are raised to powers. Some important properties include:
  • Product of powers: When multiplying numbers with the same base, add their exponents. For instance, \(a^m \times a^n = a^{m+n}\).
  • Power of a power: When taking a power of a power, multiply the exponents: \((a^m)^n = a^{m \cdot n}\).
  • Power of a product: To distribute an exponent to a product inside parentheses, such as \((ab)^n = a^n \times b^n\).
In our exercise, since the cube roots have different bases (4 and 16), we first multiply the base numbers and then compute the cube root of the result: \((4 \times 16)^{1/3} = 64^{1/3}\).
Simplifying by Multiplying Radicals
Multiplying radicals involves two main steps: multiply the numbers inside the radicals and then simplify the resulting radical. Let's revisit the steps and see how they work:
  • Begin by multiplying the radicands (the numbers inside the radical signs). For \(\sqrt[3]{4} \times \sqrt[3]{16}\), we multiply 4 and 16 to get 64.
  • Next, simplify the expression by finding the new cube root. Since \(64 = 4^3\), the cube root of 64 is 4. \(\sqrt[3]{64} = 4\).
These steps allow us to simplify the radical expression effectively and validate why our final result from the exercise is 4. Understanding the multiplication and simplification of radicals aids in transforming complex radical expressions into more understandable and manageable forms.

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Most popular questions from this chapter

Use scientific notation and the laws of exponents to perform the indicated operations. Give the result in scientific notation rounded to two significant figures. $$(0.00000003)(6000000000000)$$

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Evaluate each absolute value expression. $$|7-11|$$

Use interval notation to represent the subset of real numbers that is indicated by the inequality. $$-4
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