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Chapter 8: Problem 90

Evaluate. $$\sqrt[3]{64}$$

Short Answer

Expert verified
The cube root of 64 is 4.

Step by step solution

01

Identify the Problem

You need to evaluate the cubic root of 64, which is written as \(\frac{64}{3}\).
02

Rewrite the Expression

The problem requires finding the number that, when cubed, equals 64. So, you are looking for \(\frac{x}{3}: x^3 = 64\).
03

Find the Cube Root

Determine the value of \(x\) by finding a number that satisfies \(x^3 = 64\). By testing different values, you find that \(4^3 = 4 \times 4 \times 4 = 64\), so \(x = 4\).
04

State the Final Answer

Thus, the cube root of 64 is \(4\).

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Key Concepts

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

Evaluate Expressions
In algebra, evaluating expressions means finding the value of an expression for a given variable. For the given problem, we need to evaluate \( \sqrt[3]{64} \). This means we want to find the value that, when multiplied by itself three times, equals 64.

For simpler problems, we can use basic arithmetic. For more complex ones, we often rely on algebraic manipulation and the properties of exponents.

Evaluating expressions accurately is a key skill in algebra. It helps you solve equations and understand more advanced topics.
Cubic Roots
Cubic roots are a fundamental concept in mathematics. The cubic root of a number is the value that, when cubed (multiplied by itself twice more), gives the original number.

In our example, we need to determine \( \sqrt[3]{64} \). Think of it as finding a number \( x \) such that \( x^3 = 64 \).

Cubic roots are especially important because they're used in solving higher-degree polynomials and complex algebraic equations.

By knowing that \( 4^3 = 4 \times 4 \times 4 = 64 \), we see that \( \sqrt[3]{64} = 4 \). Understanding cubic roots can make other math topics less intimidating.
Algebraic Operations
Algebraic operations are actions we perform on algebraic expressions, including addition, subtraction, multiplication, division, and finding roots.

In the example of evaluating \( \sqrt[3]{64} \), we are using division in a unique way. Rather than dividing two numbers directly, we are finding a value that, when raised to the power of 3, equals 64.

Mastering these basic operations allows students to solve a variety of algebraic problems, making more complex equations simpler to handle.

These skills are crucial for progressing in math, especially when dealing with polynomial equations and other advanced topics.

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Most popular questions from this chapter

Simplify. $$\sqrt[3]{4} \sqrt[3]{24}+\sqrt[3]{12}$$

Simplify. $$\sqrt[3]{-1}$$

Evaluate. $$\sqrt[3]{-64}$$

The length of a rectangle is \(3 \mathrm{ft}\) more than twice its width. The perimeter of the rectangle is \(66 \mathrm{ft}\). Use a system of linear equations to find its length and width.

Simplify. $$\sqrt[3]{24}-\sqrt[3]{81}$$
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