Problem 20 - College Algebra with Modeling and Visualization

Question

Find the cube root of the number. $$ -\frac{1}{64} $$

Step-by-Step Solution

Answer

The cube root of \(-\frac{1}{64}\) is \(-\frac{1}{4}\).

Step 1: Understand Cube Roots

A cube root of a number \( x \) is a number \( a \) such that \( a^3 = x \). This means we need to find a number \( a \) for which \( a^3 = -\frac{1}{64} \).

Step 2: Analyze the Number

The given number is \(-\frac{1}{64}\), which is a negative fraction. We'll need to find the cube root of both the numerator (-1) and the denominator (64).

Step 3: Cube Root of Numerator

The numerator is \(-1\). The cube root of \(-1\) is \(-1\), since \((-1)\times(-1)\times(-1) = -1\).

Step 4: Cube Root of Denominator

The denominator is 64. The cube root of 64 is 4, since \(4 \times 4 \times 4 = 64\).

Step 5: Combine the Results

Combine the results from the numerator and denominator. The cube root of \(-\frac{1}{64}\) becomes \(-\frac{1}{4}\).

Step 6: Verify the Solution

To verify, cube the result \(-\frac{1}{4}\): \[ \left(-\frac{1}{4}\right)^3 = \left(-1\right)^3 \times \left(\frac{1}{4}\right)^3 = -\frac{1}{64} \]. The result is correct.

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction where both the numerator and the denominator are integers, and the denominator is not zero. Examples include numbers like \(\frac{1}{2}\), \(-3\), and even \(\frac{-7}{8}\). In the exercise given, \(-\frac{1}{64}\) is a rational number because it can be written in the form of a fraction.

• Numerator: -1
• Denominator: 64

Rational numbers also include whole numbers and integers, because any whole number \( n \) can be rewritten as \( \frac{n}{1} \). When dealing with cube roots, a rational number like \(-\frac{1}{64}\) requires us to find cube roots of both the numerator and the denominator separately. This approach simplifies the process of finding cube or other roots for rational numbers.

Negative Numbers

Negative numbers are simply numbers less than zero, represented by a minus sign \((-\)). When dealing with cube roots, having a negative number can impact the solution because taking an odd root of a negative number will itself result in a negative number. In our exercise, the number \(-\frac{1}{64}\) is negative.

• This indicates the cube root will also be negative.
• Cubing \(-1\) results in \(-1\), meaning the cube root of \(-1\) is \(-1\).

Thus, when solving the cube root of \(-\frac{1}{64}\), the negativity of the number remains integral to finding the correct solution.

Fractional Exponents

Fractional exponents are a way to represent roots as exponents. The cube root of a number \( x \) can be written as \( x^{1/3} \). This approach is especially useful in simplifying calculations that involve roots. The concept allows us to apply exponent rules more easily to fractions. For the exercise \(-\frac{1}{64}\), knowing that the expression can be rewritten using fractional exponents can simplify understanding, as we see:

• The base is \(-\frac{1}{64}\).
• It is represented as \((-\frac{1}{64})^{1/3} = -\frac{1}{4}\).

These fractional forms provide a clear pathway to applying consistent mathematical rules, especially when verifying solutions like cubing \(-\frac{1}{4}\) to return to \(-\frac{1}{64}\).