Problem 39 - Elementary and Intermediate Algebra

Question

Simplify. Do not use negative exponents in the answer. \(-8^{-2}\)

Step-by-Step Solution

Answer

The simplified form is \( \frac{1}{64} \).

Step 1: Understanding Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the absolute value of the exponent. In this case, \(-8^{-2} \) becomes \( \frac{1}{8^{2}}\).

Step 2: Calculate the Exponent

Now, compute the value of \(8^{2}\), which is \(8 \times 8 = 64\).

Step 3: Formulate the Reciprocal

Apply the reciprocal to the base with positive exponent: \( \frac{1}{8^2} = \frac{1}{64}\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the "base," to the power of an exponent. This operation is often written as \( x^n \), where \( x \) is the base and \( n \) is the exponent. The exponent tells us how many times the base is multiplied by itself. For example, \( 8^2 \) means 8 is multiplied by itself to get \( 8 \times 8 = 64 \).
When dealing with positive exponents, the operation is straightforward since it simply involves repeated multiplication. However, negative exponents require us to take a slight detour in our calculations, which we'll discuss soon. It's essential to grasp the behavior of exponentiation both with positive and negative exponents to simplify expressions correctly.

Reciprocal

The reciprocal of a number is essentially 1 divided by that number. It flips the number on its head, and when the number and its reciprocal are multiplied, they yield 1. For example, the reciprocal of 8 is \( \frac{1}{8} \), because \( 8 \times \frac{1}{8} = 1 \).
When dealing with negative exponents, the concept of the reciprocal plays a crucial role. A negative exponent, such as in \( 8^{-2} \), is a signal to flip the base into its reciprocal and raise the result to the positive power of the exponent. Thus, \( 8^{-2} \) becomes \( \frac{1}{8^2} \) or \( \frac{1}{64} \). By understanding how reciprocals work with exponents, you can easily transform expressions with negative exponents into simpler forms.

Simplifying Expressions

Simplifying expressions involves reducing complex mathematical expressions into their simplest form. This makes them easier to understand and work with, especially in algebraic calculations.
When your task is to simplify expressions with exponents, especially those involving negative exponents, follow these steps:

• Convert any negative exponents by turning them into reciprocals.
• Calculate the value of the positive exponent.
• Simplify the resulting fractions if possible.

So, for \(-8^{-2}\), first, we rewrite it as \( \frac{1}{8^2} \). Then, we compute \( 8^2 \) to get 64. Finally, express it as \( \frac{1}{64} \). By learning to simplify effectively, you can handle complex problems with ease and confidence.