Problem 68 - Elementary and Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. See Example 7. $$ 8^{-1 / 3} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{1}{2}\).

Step 1: Understand the Meaning of the Expression

We are given the expression \(8^{-1/3}\). This is represented as a power expression where 8 is the base, and \(-1/3\) is the exponent.

Step 2: Apply Negative Exponent Rule

A negative exponent indicates that we can take the reciprocal of the base. Thus, \(8^{-1/3} = \frac{1}{8^{1/3}}\). Now, we need to focus on calculating \(8^{1/3}\).

Step 3: Evaluate the Cube Root

The expression \(8^{1/3}\) implies finding the cube root of 8. Since \(2^3 = 8\), \(8^{1/3} = 2\).

Step 4: Simplify the Expression

Replace \(8^{1/3}\) with 2 in the reciprocal expression. Thus, \(\frac{1}{8^{1/3}} = \frac{1}{2}\).

Step 5: Final Answer

The simplified expression of \(8^{-1/3}\) is \(\frac{1}{2}\).

Key Concepts

Negative Exponents

In mathematics, a negative exponent is a way to signify the reciprocal of a positive exponent. When you see an expression like \(8^{-1/3}\), the negative exponent means that you first take the reciprocal of the base number raised to the positive version of that exponent.

Here's a step-by-step look at how to handle negative exponents:

• Identify the base and the negative exponent. For example, in \(8^{-1/3}\), 8 is the base, and \(-1/3\) is the exponent.
• Convert the expression to its reciprocal by changing the negative exponent into a positive. Use the formula: \(a^{-m} = \frac{1}{a^m}\). So, \(8^{-1/3}\) becomes \(\frac{1}{8^{1/3}}\).

This conversion simplifies the problem, allowing you to focus on solving the expression with a positive exponent.

Cube Roots

Cube roots are the opposite operation of cubing a number. To cube a number means to multiply it by itself twice, whereas a cube root aims to find a number that, when cubed, gives the original number. For example, when dealing with \(8^{1/3}\):

To find the cube root of a number:

• Identify the number you need the cube root of, which is the base in our expression \(8^{1/3}\).
• Think of the number that when multiplied by itself three times yields the base number. So, since \(2 \times 2 \times 2 = 8\), \(2^3 = 8\), this means the cube root of 8 is 2.

Cube roots are helpful in simplifying expressions involving fractional exponents. Recognizing this relationship speeds up calculation and simplification.

Reciprocals

Understanding reciprocals is crucial when working with expressions involving inverse operations. A reciprocal of a number is essentially 1 divided by that number. For example, the reciprocal of 2 is \(\frac{1}{2}\).

When you encounter a negative exponent like in \(8^{-1/3}\), the concept of reciprocals helps transform the expression:

• The negative exponent \(-1/3\) initially suggests that 8 should be handled as a reciprocal, resulting in \(\frac{1}{8^{1/3}}\).
• Once you evaluate the cube root of 8 to find it equals 2, you replace and simplify the expression to \(\frac{1}{2}\).

Reciprocals simplify expressions and allow us to handle negative exponents intuitively by flipping the fractions.