Problem 32 - Elementary and Intermediate Algebra

Question

Express using positive exponents and simplify, if possible. \(7^{-1}-7^{0}\)

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{-6}{7}\).

Step 1: Understand Negative Exponent

The expression includes a term with a negative exponent, which is \(7^{-1}\). To express this with a positive exponent, recall that \(a^{-n} = \frac{1}{a^n}\). So, \(7^{-1} = \frac{1}{7}\).

Step 2: Simplify Zero Exponent

Look at the second term \(7^{0}\). Any non-zero number raised to the power of 0 is 1. So, \(7^{0} = 1\).

Step 3: Substitute Simplified Terms

Substitute the simplified values back into the expression. The original expression \(7^{-1} - 7^{0}\) becomes \(\frac{1}{7} - 1\).

Step 4: Common Denominator

To simplify \(\frac{1}{7} - 1\), find a common denominator, which is 7. Rewrite 1 as \(\frac{7}{7}\) so both terms have the same denominator: \(\frac{1}{7} - \frac{7}{7}\).

Step 5: Subtract Fractions

Subtract the numerators of the fractions while keeping the denominator the same: \(\frac{1}{7} - \frac{7}{7} = \frac{1 - 7}{7} = \frac{-6}{7}\).

Key Concepts

Positive Exponents

When working with exponents, moving from negative to positive is crucial for simplification. A positive exponent describes repeated multiplication. For instance, the term \(7^2\) means \(7\) multiplied by itself, which is \(7 \times 7 = 49\). To transform a negative exponent into a positive one, use the rule \(a^{-n} = \frac{1}{a^n}\). This flips the base to the denominator. For example, \(7^{-1}\) becomes \(\frac{1}{7}\). This transition is essential for simplifying expressions and making calculations more intuitive. Remember:

• Positive exponent multiplies the base number repeatedly.
• Negative exponent turns into a reciprocal.
• Use this property to simplify negative exponent expressions.

Zero Exponent

The concept of a zero exponent is straightforward yet powerful. Any non-zero number raised to the zero power always equals one. Mathematically, if \(aeq 0\), then \(a^0 = 1\). This rule applies universally, simplifying many expressions. Even when calculations present complex bases, the zero exponent reduces the result drastically.
For example, in the expression \(7^0\), it's tempting to calculate, but instead, note the result is 1. This property holds for all non-zero bases:

• \(7^0 = 1\)
• \(10^0 = 1\)
• \(10000^0 = 1\)

Realizing this can save time and prevent unnecessary calculations, making zero exponents a valuable simplification tool.

Fraction Subtraction

Subtraction of fractions is a fundamental arithmetic operation. It becomes necessary to ensure the fractions have like denominators before subtraction. When subtracting fractions such as \(\frac{1}{7} - 1\), we must first rewrite 1 with a denominator matching the other fraction. Thus, 1 is the same as \(\frac{7}{7}\). With common denominators, the subtraction process involves subtracting numerators while maintaining the denominator.
For the expression becomes:

• Convert \(1\) to \(\frac{7}{7}\), making \(\frac{1}{7} - \frac{7}{7}\).
• Subtract the numerators: \(1 - 7 = -6\).
• The resulting fraction is \(\frac{-6}{7}\).

This method ensures accuracy and simplicity, allowing easy management of more complex fractional operations.