Problem 21 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ a^{-8} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \(a^{-8}\) with a positive exponent. Answer: \(\frac{1}{a^8}\)

Step 1: Apply the negative exponent rule

According to the rule for negative exponents, \(a^{-n} = \frac{1}{a^n}\). So, we will apply this rule to rewrite the given expression, \(a^{-8}\), as a fraction with a positive exponent.

Step 2: Rewrite the expression with a positive exponent

Using the rule for negative exponents, we can rewrite \(a^{-8}\) as: $$ a^{-8} = \frac{1}{a^8} $$ Now, the expression is written with only positive exponents.

Key Concepts

Positive Exponents

When working with positive exponents, we're dealing with the more conventional terms that people are familiar with in mathematics. Positive exponents relay the basic idea of multiplication repeated. For instance, the expression \(a^3\) means that we multiply the base \(a\) by itself three times, resulting in \(a \times a \times a\).

Understanding positive exponents is fundamental as they reflect the number of times a number or variable is used as a factor in a multiplication. They are straightforward to compute, as long as one remembers that \(a^1 = a\) and \(a^0 = 1\), regardless of the value of \(a\), provided that \(a\) is nonzero. It's essential to stress the fact that exponents are just a succinct way to express long multiplications.

Exponent Rules

To effectively work with exponents, certain rules or laws of exponents should be utilized. These rules help in simplifying and evaluating algebraic expressions efficiently. One of the primary rules is the negative exponent rule, which states that for any nonzero number \(a\) and positive integer \(n\), the expression \(a^{-n}\) equals \(\frac{1}{a^n}\).

Other vital exponent rules include:

• The product rule: \(a^m \times a^n = a^{m+n}\).
• The quotient rule: \(\frac{a^m}{a^n} = a^{m-n}\), assuming \(a\) is not zero and \(m\) is greater than \(n\).
• The power of a power rule: \((a^m)^n = a^{mn}\).
• The power of a product rule: \((ab)^n = a^n b^n\).

These rules serve as the foundation for manipulations involving exponents and are particularly helpful when dealing with more complex algebraic expressions.

Algebraic Expressions

In the world of algebra, algebraic expressions are the phrases and sentences. An algebraic expression is a combination of variables, numbers, and at least one arithmetic operation. For example, \(4x^2 - 3x + 7\) is an algebraic expression where \(x\) is the variable. These expressions can be as simple as a single term, like \(a^3\), or as complex as polynomials with many terms, such as \(3x^2 + 2x - 5\).

Algebraic expressions become especially interesting and challenging when we apply various arithmetic operations and combine them with exponent rules. Simplifying these expressions involves combining like terms, understanding the order of operations, and using exponent rules effectively. One can always expect to encounter algebraic expressions in mathematical studies, and mastering their manipulations is a critical skill in algebra.