Problem 22 - Elementary and Intermediate Algebra

Question

Express using positive exponents and simplify, if possible. \(7^{-2}\)

Step-by-Step Solution

Answer

The expression \(7^{-2}\) simplifies to \(\frac{1}{49}\).

Step 1: Understand Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent. For example, given the expression \( a^{-n} \), it represents the reciprocal of \( a^n \). In our case, the base is 7, and the exponent is -2.

Step 2: Rewrite with Positive Exponents

Apply the rule for negative exponents. The expression \( 7^{-2} \) becomes \( \frac{1}{7^2} \). This conversion allows us to work with positive exponents.

Step 3: Calculate the Positive Exponent

Now calculate \( 7^2 \). This means multiplying the base, 7, by itself: \( 7 \times 7 = 49 \).

Step 4: Simplify the Expression

Replace \( 7^2 \) in the reciprocal: \( \frac{1}{49} \). The expression is now fully simplified using positive exponents.

Key Concepts

Positive Exponents

Positive exponents are handy tools in mathematics that indicate how many times a base is used as a factor. In simple terms, a positive exponent tells you to multiply the base by itself. For example, with the expression \(7^2\), the base is 7, and the exponent is 2. This means 7 is multiplied by itself: \(7 \times 7 = 49\).

Positive exponents always result in larger numbers when the base is greater than 1. They help simplify complex expressions and make calculations easier by allowing us to see the growth of numbers. Working with positive exponents also prepares us for more advanced concepts, like polynomial multiplication or even scientific notation.

Reciprocals

Reciprocals are essential in understanding negative exponents. A reciprocal simply flips the numerator and the denominator of a fraction. For example, the reciprocal of 3 is \(\frac{1}{3}\), and vice versa.

When it comes to exponents, a negative exponent means that we find the reciprocal of the base raised to the corresponding positive exponent. For instance, \(7^{-2}\) turns into \(\frac{1}{7^2}\). Here, the negative sign indicates we need to flip the fraction, essentially transforming it into a fraction consisting of its reciprocal.

This concept is vital, especially when simplifying expressions, as it allows us to deal comfortably with negative powers and transform them into more workable positive exponent expressions.

Simplifying Expressions

Simplifying expressions is a fundamental step in mathematics that helps us make sense of complex problems. It's all about reducing expressions to their simplest form while ensuring they remain equivalent.

Starting with basic operations, we aim to combine like terms and perform arithmetic efficiently. For example, \(7^{-2}\), when simplified, becomes \(\frac{1}{49}\) after rewriting using positive exponents and calculating that \(7^2 = 49\).

This process often involves using positive exponents, identifying and rewriting reciprocals, and eventually following through necessary calculations. By simplifying expressions, we make them more understandable and more manageable, allowing us to solve equations faster and with less room for error. It’s like de-cluttering a room: the cleaner it is, the easier it is to find what you need.