Problem 40 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ a^{7} b^{-9} z w^{3} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \(a^7 b^{-9} z w^3\) using only positive exponents. Answer: \(\frac{a^7z w^3}{b^9}\)

Step 1: Identify terms with negative exponents

In the given expression \(a^7 b^{-9} z w^3\), the term with a negative exponent is \(b^{-9}\). We will rewrite this term using the rule \(a^{-n} = \frac{1}{a^n}\).

Step 2: Rewrite the expression with only positive exponents

Applying the exponent rule to \(b^{-9}\) gives us \(\frac{1}{b^9}\). Now we rewrite the entire expression, replacing \(b^{-9}\) with \(\frac{1}{b^9}\), keeping all other terms the same: $$ a^7 b^{-9} z w^3 = a^7 \frac{1}{b^9} z w^3 $$

Step 3: Simplify the expression

The expression is now in the form with only positive exponents, and is simplified as: $$ a^7 \frac{1}{b^9} z w^3 = \frac{a^7z w^3}{b^9} $$

Key Concepts

Negative Exponents

Understanding negative exponents is crucial when dealing with algebraic expressions. In essence, a negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. For example, when we see something like \(x^{-n}\), it's equivalent to \(\frac{1}{x^n}\). This reciprocity means that you are essentially dividing by the base number \(n\) times.

Applying this concept to simplify algebraic expressions can reduce complexity and ensure the expression contains only positive exponents, which are often easier to work with. Remember, the base with a negative exponent never equals zero; it's defined only for non-zero values.

Simplifying Expressions

Simplifying expressions with exponents often involves applying a set of rules systematically to reduce the expression to its most basic form. The goal is to make the expression easier to understand or prepare it for further mathematical operations like solving or factoring. Simplification may involve combining like terms, using the distributive property, and managing different types of exponents by turning any negative exponents into positive ones.

When simplifying, it is also essential to maintain the equivalence of the original expression. It means that whatever manipulations we apply to the expression, the value should remain unchanged. This process can involve several steps, but with practice, it becomes a straightforward and useful skill in algebra.

Exponent Rules

A variety of exponent rules govern the operations involving powers. The most fundamental of these include the Product Rule, Quotient Rule, Power Rule, and the rules for dealing with zero and negative exponents. These rules are vital tools to correctly manipulate and simplify expressions involving exponents.

The Product Rule states that when multiplying two powers with the same base, you add the exponents (\(a^m \cdot a^n = a^{m+n}\)). Conversely, the Quotient Rule dictates that when dividing two powers with the same base, you subtract the exponents (\(\frac{a^m}{a^n} = a^{m-n}\)). The Power Rule says that when raising a power to another power, you multiply the exponents (\((a^m)^n = a^{m \cdot n}\)). And for negative exponents, we know that \(a^{-n} = \frac{1}{a^n}\) as long as \(a eq 0\).

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operation symbols that represent mathematical relationships. They can include terms, factors, coefficients, powers, and can be as simple as a single term or as complex as a long phrase of diverse parts.

Manipulating these expressions requires an understanding of algebra's fundamental principles and the ability to apply the appropriate exponent rules. It's the language through which we can express general mathematical ideas and find unknown values. Simplifying algebraic expressions, which includes making sure all exponents are positive, is like cleaning up the expression to remove any distractions and reveal its core components.