Problem 58 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ 7 a\left(a^{2}-4\right)^{-2}\left(b^{2}-1\right)^{-2} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \(7a\left(a^{2}-4\right)^{-2}\left(b^{2}-1\right)^{-2}\) using only positive exponents. Answer: \(\frac{7a}{\left(a^{2}-4\right)^{2}\left(b^{2}-1\right)^{2}}\)

Step 1: Identify the terms with negative exponents

In the given expression, we have two terms with negative exponents: \(\left(a^{2}-4\right)^{-2}\) and \(\left(b^{2}-1\right)^{-2}\). These are the terms we need to rewrite with positive exponents.

Step 2: Rewrite terms with negative exponents as reciprocals with positive exponents

We can rewrite each term with a negative exponent as its reciprocal with a positive exponent. For the term \(\left(a^{2}-4\right)^{-2}\), we can rewrite it as: $$ \frac{1}{\left(a^{2}-4\right)^{2}} $$ For the term \(\left(b^{2}-1\right)^{-2}\), we can rewrite it as: $$ \frac{1}{\left(b^{2}-1\right)^{2}} $$

Step 3: Combine the simplified terms

Now we substitute the simplified terms back into the original expression: $$ 7 a\left(a^{2}-4\right)^{-2}\left(b^{2}-1\right)^{-2} = 7a \cdot \frac{1}{\left(a^{2}-4\right)^{2}} \cdot \frac{1}{\left(b^{2}-1\right)^{2}} $$ We can combine the fractions and obtain the final expression: $$ \frac{7a}{\left(a^{2}-4\right)^{2}\left(b^{2}-1\right)^{2}} $$ Now we have written the given expression with only positive exponents.

Key Concepts

Negative Exponents

Understanding negative exponents is crucial in algebra as they frequently appear in various mathematical contexts. The vital thing to remember is that a negative exponent indicates that the number or the expression is on the 'wrong side' of a fraction and needs to be inverted. For example, consider the expression \( x^{-n} \). To transform this into an expression with a positive exponent, you would take the reciprocal of the base and then apply the positive exponent, making it \( \frac{1}{x^n} \).

Applying this concept to algebraic expressions like \( (a^2 - 4)^{-2} \) from the exercise, we invert the whole expression to \( \frac{1}{{(a^2 - 4)}^2} \), effectively changing the negative exponent into a positive one. This inversion is crucial for simplifying expressions and solving equations.

In essence, negative exponents prompt us to evaluate the reciprocal of the base raised to the corresponding positive power, a relatively straightforward yet powerful mathematical concept.

Reciprocal Notation

Reciprocal notation is a way to express one number as the inverse of another. In basic terms, the reciprocal of a number \( x \) is \( \frac{1}{x} \). It's important to note that the reciprocal of a fraction is simply the inverted fraction, e.g., the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \).

When working with negative exponents as shown in the exercise, reciprocal notation becomes especially handy. The term \( (b^2 - 1)^{-2} \) is expressed with a reciprocal as \( \frac{1}{{(b^2 - 1)}^2} \). This transformation allows for positive exponentiation within the framework of standard arithmetic operations.

Being adept with reciprocal notation makes it easier to understand and work with fractions, division, and concepts related to inverse operations, a foundation for more complex algebraic manipulations.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that represent a specific value. They are the building blocks of algebra and are used to describe patterns, construct functions, and solve problems. An understanding of how to manipulate these expressions is fundamental to success in algebra.

In the given exercise, we see an algebraic expression \( 7a(a^2 - 4)^{-2}(b^2 - 1)^{-2} \). This expression comprises variables (\( a \) and \( b \)), constants (7 and the numbers within the parentheses), and operations (multiplication and exponentiation). The goal is to rewrite it with positive exponents, a common task that requires the use of reciprocal notation for negative exponents.

Once students master the mechanics of manipulating algebraic expressions—like changing the signs of exponents, combining like terms, and evaluating operations in the correct order—they can tackle a wide range of algebraic problems and develop deeper mathematical understanding.