Problem 82 - Introductory Algebra for College Students

Question

Evaluate: \(125^{-4}\). (Section 8.6, Example 3)

Step-by-Step Solution

Answer

\(125^{-4}\) equals \(1 / 244140625\).

Step 1: Identify the Base and Exponent

The base here is 125 and the exponent is -4. The negative sign in the exponent tells us that the base (125) needs to be in the denominator of a fraction.

Step 2: Apply the Negative Exponent Rule

The negative sign in the exponent is resolved by taking the reciprocal of the base. So, \(125^{-4}\) becomes \(1 / 125^4\).

Step 3: Calculate the Denominator

We calculate \(125^4\) which is \(125 * 125 * 125 * 125\). This results in a value of 244140625.

Step 4: Write the Final Answer

Therefore, \(125^{-4}\) equals \(1 / 244140625\).

Key Concepts

negative exponents

Negative exponents may seem a little intimidating at first, but they're not as complex as they appear. In essence, a negative exponent indicates that you need to take the reciprocal of the base.

The negative sign in the exponent doesn't affect the base itself; instead, it tells us to "flip" the base into the denominator.

Let's consider an example:

• For any base number, such as 125, with a negative exponent like -4, you first take the reciprocal, turning it into a fraction: \[ 125^{-4} = \frac{1}{125^4} \]

Remember, the base remains the same, and only the position changes from numerator to denominator.

This process is a key part of simplifying expressions using negative exponents.

reciprocal

The concept of a reciprocal is crucial when working with negative exponents. A reciprocal simply means flipping a number; if it's a fraction, you swap the numerator and the denominator.

For whole numbers, like 125, its reciprocal is expressed as a fraction:

• Consider 125, which is equivalent to \( \frac{125}{1} \). Its reciprocal becomes \( \frac{1}{125} \).

Reciprocals are an efficient way to simplify expressions by helping to get rid of negative exponents.

When you have a term like \( 125^{-4} \), taking the reciprocal allows the exponent to turn positive after moving the base to the denominator:

• Change \( 125^{-4} \) into \( \frac{1}{125^4} \).

Remember to perform operations like multiplication afterward to find your final result.

base and exponent identification

Identifying the base and its corresponding exponent is the first step in solving any expression involving exponents, whether they're positive or negative.

The base is the number that is repeatedly multiplied, and the exponent indicates how many times to multiply it by itself.

• In our example, the base is 125, and the exponent is -4.

The exponent, when negative, not only controls the multiplication but also signals the need for calculating a reciprocal.

Once the base and exponent are identified:

• Apply the rules of exponentiation while being mindful of the sign of the exponent.
• Use the base with its reciprocal when the exponent is negative.

This foundational skill sets the stage for more complex calculations and helps avoid errors in solving problems like \( 125^{-4} \).