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Chapter 9: Problem 26

Evaluate each radical without using a calculator or a table. (Objective 1) $$\sqrt[3]{64}$$

Short Answer

Expert verified
The cube root of 64 is 4.

Step by step solution

01

Understand the expression

We are given the expression \( \sqrt[3]{64} \). This is asking for the cube root of 64, which means finding a number that, when multiplied by itself three times (cubed), gives 64.
02

Determine the cube root

Think about the numbers that, when cubed, could yield 64. Start with smaller numbers:- \( 2^3 = 2 \times 2 \times 2 = 8 \)- \( 3^3 = 3 \times 3 \times 3 = 27 \)- \( 4^3 = 4 \times 4 \times 4 = 64 \)You can see that the number 4, when cubed, equals 64.
03

Verify the solution

We determined that 4 is the cube root of 64, since \( 4^3 = 64 \). To verify:- Calculate \( 4^3 = 4 \times 4 = 16 \), and then \( 16 \times 4 = 64 \), which confirms that \( \sqrt[3]{64} = 4 \).

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Key Concepts

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

Understanding Radicals
Radicals allow us to express roots, like square roots and cube roots, in mathematical form. When dealing with cube roots, we are looking for a value that, when multiplied by itself three times, equals the given number. This is known as the radicand.
For example, in the expression \( \sqrt[3]{64} \), 64 is the radicand, and we are searching for a number (let's call it \( x \)) such that \( x \times x \times x = 64 \).
Understanding radicals helps break down complex expressions into more manageable parts, making calculation and simplification easier. However, to find cube roots, especially without a calculator, requires familiarity with numbers and their patterns.
Evaluation Without Using a Calculator
Evaluating expressions like cube roots without a calculator relies on basic arithmetic skills and number sense. Let's explore how you can accomplish this using our example of \( \sqrt[3]{64} \).
The most efficient way is to start with simple integers and consider their cubes:
  • \( 2^3 = 2 \times 2 \times 2 = 8 \)
  • \( 3^3 = 3 \times 3 \times 3 = 27 \)
  • \( 4^3 = 4 \times 4 \times 4 = 64 \)
By evaluating these steps, it's clear that 4 is the cube root of 64. Practicing this method with various numbers can develop your ability to estimate and evaluate roots accurately.
Tip: Knowing the cubes of the first few integers by heart can greatly speed up this process.
The Role of Mathematical Reasoning
Mathematical reasoning is a critical skill in evaluating roots without directly using computational tools. It's about thinking logically through a problem and finding solutions step-by-step.
Here, let's apply reasoning to the cube root of 64:
  • We need a number whose cube equals 64.
  • Recognizing that \( 4^3 \) gives us 64 confirms that 4 is indeed the answer.
Verification plays a crucial role, and in this case, we checked the calculation by cubing 4:
  • \( 4 \times 4 = 16 \)
  • \( 16 \times 4 = 64 \)
This confirms our reasoning and solidifies understanding.
Using mathematical reasoning encourages deeper comprehension of numbers and their relationships, building confidence for tackling more complex math challenges.

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Most popular questions from this chapter

Use a calculator to find a rational approximation of each square root. Express your answers to the nearest hundredth. $$\sqrt{75}$$

Simplify each expression. $$3 \sqrt{8}-5 \sqrt{20}-7 \sqrt{18}-9 \sqrt{125}$$

For Problems \(43-54\), find each product and express your answers in simplest radical form. All variables represent nonnegative real numbers. $$ \sqrt[3]{25 x^{2}} \sqrt[3]{5 x} $$

Express each radical in simplest radical form. All variables represent nonnegative real numbers. $$\sqrt{28 x^{3} y}$$

Use a calculator to find a rational approximation of each square root. Express your answers to the nearest hundredth. $$\sqrt{50}$$
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