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Chapter 8: Problem 51

Find each cube root. $$-\sqrt[3]{8}$$

Short Answer

Expert verified
The cube root of -8 is -2

Step by step solution

01

Identifying the cube

Here the number within the cube root is 8 which is a perfect cube number. It can be written as 2*2*2 = \(2^3\).
02

Taking the cube root

The cube root of 2*2*2 or \(2^3\) is 2. As the number was under the cube root was negative, the cube root will also be negative.

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Key Concepts

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

Perfect Cube
A perfect cube is a number that can be formed by multiplying an integer by itself twice. This means it takes the shape of \( n \times n \times n \), where \( n \) is an integer.

When you think of a perfect cube, try imagining a small cube made of smaller unit cubes. Each side of this large cube has the same number of unit cubes. For instance, the number 8 is a perfect cube because it can be written as \( 2^3 \), or \( 2 \times 2 \times 2 \).

When dealing with cube roots, identifying whether the number is a perfect cube can simplify the process.
  • If a number can be expressed in the form of some integer raised to the power of 3, it鈥檚 a perfect cube.
  • Taking the cube root of a perfect cube yields a whole number, which makes calculations straightforward.
Negative Cube Root
Calculating the cube root of a negative number can seem tricky initially, but is actually quite simple. A negative cube root is simply the opposite of the positive cube root for perfect cubes.

When you see a negative sign outside the cube root, it indicates that the result will also be negative. For instance, take the cube root of -8: the cube root of a positive 8 is 2, and therefore, the cube root of -8 is -2.

This means:
  • The cube root of any negative number will itself be negative.
  • Since \( (-2)^3 = -8 \), this confirms that the cube root of -8 is indeed -2.
Using negative cube roots is especially useful in equations where you need to maintain sign consistency across calculations.
Exponents
The concept of exponents is crucial in understanding cube roots and perfect cubes. An exponent denotes how many times a number, known as the base, is multiplied by itself.

For example, in \( 2^3 \), the number 2 is the base, and the 3 is the exponent. This expression equates to \( 2 \times 2 \times 2 = 8 \).

When calculating cube roots, you reverse the process of exponents. If \( n^3 = 8 \), then \( n \) is the cube root of 8, which is 2.
  • Exponents simplify repeated multiplication processes.
  • Understanding them helps in breaking down and simplifying expressions within cube roots, especially in algebra.
Getting comfortable with exponents allows you to easily switch between exponential and root forms, which is essential for problems involving powers and roots.

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Most popular questions from this chapter

The formula $$A=1.83 w^{\frac{2}{3}}$$ models the area of an egg shell, \(A\), in square centimeters, in terms of its weight, \(w,\) in grams. a. An ostrich egg weighs approximately 1600 grams. Use a calculator to approximate the shell's area to the nearest tenth of a square centimeter. b. Rewrite the formula in radical notation.

In Exercises \(53-74\), rationalize each denominator. Simplify, if possible. $$\frac{18}{3-\sqrt{3}}$$

What is the meaning of \(a^{-\frac{m}{n}} ?\) Give an example.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. When I use the definition for \(a^{\frac{m}{n}},\) I prefer to first raise \(a\) to the \(m\) power because smaller numbers are involved.

Without using a calculator and knowing that \(\sqrt{2} \approx 1.4142\) rationalizing the denominator of \(\frac{1}{\sqrt{2}}\) makes division to obtain a decimal approximation for \(\frac{1}{\sqrt{2}}\) easier to perform. Because 10 and 8 share a common factor of \(2,\) I simplified \(\frac{\sqrt{10}}{8}\) to \(\frac{\sqrt{5}}{4}\)
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