Problem 22 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ a^{-10} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \(a^{-10}\) using only positive exponents. Answer: \(\frac{1}{a^{10}}\)

Step 1: Apply the negative exponent rule

Using the negative exponent rule, rewrite the expression with positive exponents: \(a^{-10} = \frac{1}{a^{10}}\)

Step 2: Simplify the expression

In this case, the expression is already in its simplest form, so there's no need to simplify further. Thus, the final answer is: \(a^{-10} = \frac{1}{a^{10}}\)

Key Concepts

Positive Exponents

Positive exponents are a fundamental concept in mathematics and represent repeated multiplication. For example, when we see an expression like \( x^3 \), it means \( x \) multiplied by itself three times: \( x \cdot x \cdot x \). Understanding positive exponents is crucial because they underpin more complex operations and are the basis from which we explore their counterparts, negative exponents.

Positive exponents also follow the law of exponents, which includes operations such as multiplying like bases by adding exponents, or raising a power to a power by multiplying exponents. Simplifying expressions with positive exponents often requires applying these exponent rules to combine or reduce terms for an easier or more compact expression.

Exponent Rules

Exponent rules are the guidelines that describe how to manipulate expressions with exponents in various mathematical operations. One primary rule is the negative exponent rule, which states that a term with a negative exponent, such as \( a^{-n} \), can be rewritten as \( \frac{1}{a^n} \). This rule allows us to transform negative exponents into positive exponents, as we see with the textbook exercise \( a^{-10} = \frac{1}{a^{10}} \).

Other crucial exponent rules include the product rule (multiplying powers with the same base), quotient rule (dividing powers with the same base), and power of a power rule (taking a power to another power). Mastering these rules enables students to simplify and solve exponential expressions efficiently, leading to a clearer understanding of exponents and their properties.

Simplifying Expressions

Simplifying expressions is a process of transforming complex algebraic expressions into simpler or more understandable forms without changing their value. This often involves applying exponent rules, combining like terms, and factoring. When simplifying expressions with exponents, one might need to convert negative exponents to positive exponents or combine terms using exponent laws.

For example, the solution offered for the exercise \( a^{-10} \) is already in its simplest form as \( \frac{1}{a^{10}} \), following the negative exponent rule. Simplifying is a critical skill in algebra that not only makes expressions easier to work with but also aids in solving equations and understanding the underlying mathematical relationships.