Problem 11 Find the $n$th root(s), if pos... [FREE SOLUTION]

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Elementary Algebra Within Reach

Ron Larson, Kimberly Nolting

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 9: Problem 11

Find the $n$th root(s), if possible. The cube root of $-8$

Short Answer

Expert verified

The cube root of $-8$ is $-2$.

Step by step solution

Identify the Cube Root

The problem asks for the cube root of $-8$, which means finding a number that when multiplied by itself twice (cubed) gives $-8$.

Solve for Cube Root

A positive number cubed gives a positive result, and a negative number cubed gives a negative result. Since we have $-8$ here, the cube root will be a negative number. Through trial and error, it can be identified that $-2$ is the cube root of $-8$, since $(-2) * (-2) * (-2) = -8$.

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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 a special value that, when multiplied by itself three times, gives the original number. For instance, finding the cube root of $-8$ involves identifying a number that, when cubed (multiplied by itself twice), results in $-8$.

To express this mathematically, if $x$ is the cube root of $-8$, then $x^3 = -8$. For cube roots, both positive and negative numbers can be considered. A negative result indicates the cube root must be a negative number.

Working with Negative Numbers

Negative numbers have unique properties, especially in operations like multiplication. When multiplying:

A positive number times a positive number yields a positive result.
A negative number times a negative number results in a positive number.
A negative number times a positive number, or vice versa, results in a negative number.

Understanding these properties is key to solving problems with negative numbers. For the cube root of $-8$, because a negative number raised to an odd power (like 3) remains negative, we know the cube root must be negative. Hence, $-2$ is the cube root because $-2 \cdot -2 \cdot -2 = -8$.

In essence, negative cube roots simplify to negative originals when cubed.

Using Trial and Error

Trial and error is a practical method for finding cube roots, especially when one is unfamiliar with mental calculations or lacks a calculator.

To find the cube root of \-8\ using trial and error:

Start with an initial guess (e.g., $ -1$ or $ -2$).
Cube the number (multiply it by itself two more times).
Adjust your guess based on whether the cube is less than or greater than $-8$.

In this problem, by trying $-2$, we compute $(-2 \times -2 \times -2 = -8$), confirming that our guess is correct.

This approach can be particularly useful in educational settings to develop intuition about numbers and powers.

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91影视

Find the \(n\)th root(s), if possible. The cube root of \(-8\)

Short Answer

Step by step solution

Identify the Cube Root

Solve for Cube Root

Key Concepts

Understanding Cube Roots

Working with Negative Numbers

Using Trial and Error

One App. One Place for Learning.

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