Problem 51 - Algebra 2 and Trigonometry

Question

In \(35-63,\) write each expression with only positive exponents and express the answer in simplest form. The variables are not equal to zero. $$ \frac{9 x^{2}}{a^{-3}} $$

Step-by-Step Solution

Answer

The expression with positive exponents is \( 9x^2 a^3 \).

Step 1: Rewrite the Denominator

Use the property of exponents that states \( a^{-n} = \frac{1}{a^n} \) to rewrite the expression. The expression becomes \( \frac{9 x^{2}}{a^{-3}} = 9x^2 \cdot a^3 \).

Step 2: Simplify the Expression

Now, the expression is \( 9x^2 \cdot a^3 \). Since there are no like terms or further multiplication necessary, this is the simplified expression.

Key Concepts

Negative Exponents

Negative exponents can seem a bit intimidating at first, but they follow straightforward rules. When you see a negative exponent, think of it as a term that needs to "move" across a fraction line to become positive. For instance, the expression \( a^{-3} \) is equivalent to \( \frac{1}{a^3} \). This means that instead of dealing with a negative exponent, you can rewrite the term in the denominator as a positive exponent.

• Negative exponents indicate reciprocal action: move the base to the opposite part of the fraction.
• They never impact the sign of the base number — only the position in fraction.

This is a critical step in simplifying expressions, helping to maintain consistency and clarity in mathematical calculations.

Simplifying Expressions

Simplifying expressions means making them as concise as possible while still maintaining their original value. In expressions containing exponents, the goal is to remove negative exponents and combine like terms using exponent rules. With our specific example, simplifying \( \frac{9 x^2}{a^{-3}} \) involves eliminating the negative exponent by rewriting it as \( 9x^2 \cdot a^3 \).Here are key points when simplifying expressions:

• Convert negative exponents into positive ones by shifting their position.
• Combine like terms wherever possible.
• Evaluate constants and simplify numeric expressions.

By carefully following these steps, complex algebraic expressions become easier to understand and use in mathematical operations.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators such as \(+\), \(-\), \(\times\), and \(\div\). They form the backbone of algebra, representing quantities whose values can change.In our example, \( 9x^2 \), \( a^{-3} \), and their simplified form \( 9x^2 \cdot a^3 \) are algebraic expressions. Mastering these expressions involves understanding:

• How to apply exponent rules to manipulate variable terms.
• The relationship between coefficients and the variables they multiply.
• The significance of each component in forming a complete expression.

Understanding algebraic expressions enables students to tackle complex problems by breaking them down into manageable parts. This foundational knowledge is crucial for higher-level algebra and calculus.