Problem 48 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ 7 a^{-2} b^{2} c^{2} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \(7 a^{-2} b^{2} c^{2}\) with positive exponents only. Answer: \(\frac{7b^2c^2}{a^2}\)

Step 1: Rewrite the expression with positive exponents

Following the exponent rule for negative exponents, we rewrite the given expression as: $$ 7 a^{-2} b^{2} c^{2} = 7 \cdot \frac{1}{a^2} \cdot b^2 \cdot c^2 $$

Step 2: Combine the terms

Now, we can combine the terms by multiplying the numerical part and the variable terms: $$ 7 \cdot \frac{1}{a^2} \cdot b^2 \cdot c^2 = \frac{7b^2c^2}{a^2} $$ The expression has been rewritten with positive exponents only: $$ \boxed{\frac{7b^2c^2}{a^2}} $$

Key Concepts

Negative Exponents Rule

When it comes to understanding algebraic expressions, the 'negative exponents rule' is crucial. Exponents, in general, indicate how many times we multiply a number by itself. But what about when you encounter an exponent that is negative, such as in the expression \(7a^{-2}b^{2}c^{2}\)? Here's the rule: a negative exponent means you take the reciprocal of the base and then apply the positive exponent. In essence, \(a^{-n}\) is the same as \(\frac{1}{a^n}\) when \(a\) is not zero. This is why the given expression \(7a^{-2}b^{2}c^{2}\) could be rewritten as \(7 \cdot \frac{1}{a^2} \cdot b^2 \cdot c^2\). By applying this rule, the students transform expressions with negative exponents into ones with only positive exponents, simplifying the problem. Remember, the base with a negative exponent must be non-zero, as division by zero is undefined.

Simplifying Algebraic Expressions

The process of simplifying algebraic expressions involves combining like terms and using the rules of arithmetic to rewrite expressions in a simpler form. This can involve expanding expressions, factoring, canceling terms, and applying exponent rules. In the example, simplifying means getting rid of the negative exponent. First, by following the rule for negative exponents, we've adjusted the expression to have only positive exponents. Then, we've combined the terms by multiplying the coefficients and variables together. This gave us \(\frac{7b^2c^2}{a^2}\), a more streamlined and easier-to-understand version of the original expression. For ease of processing, students can visualize this as separating the terms into fractions, multiplying across the numerators, and then the denominators. Simplification is all about making the expression as straightforward as possible while maintaining its original value.

Exponentiation in Algebra

Exponentiation is a form of mathematical shorthand to express repeated multiplication. It is a key component of algebra that you will encounter when working with polynomials, equations, and various functions. In the simplified expression \(\frac{7b^2c^2}{a^2}\), we see exponentiation at work with the terms \(b^2\) and \(c^2\), which tells us that \(b\) and \(c\) are each being squared, or multiplied by themselves once. Learning how to manipulate exponents is essential, as they follow a specific set of rules. For instance, when multiplying terms with the same base, you add the exponents. Conversely, when dividing, you subtract the exponents. Familiarizing oneself with these rules makes tackling algebraic problems more manageable and is fundamental to mastering more complex topics in mathematics.