Problem 28 - A Graphical Approach to College Algebra

Question

Perform the indicated operations. Write your answers with only positive exponents. Assume that all variables represent positive real numbers. $$9^{-4} \cdot 9^{-1}$$

Step-by-Step Solution

Answer

\( \frac{1}{59049} \)

Step 1: Apply the Property of Exponents for Multiplication

The property of exponents states that when multiplying like bases, you add the exponents: \[ a^m imes a^n = a^{m+n} \] In our problem, we have: \[ 9^{-4} imes 9^{-1} = 9^{-4 + (-1)} \] Now simplify the expression by adding the exponents: \[ 9^{-4 + (-1)} = 9^{-5} \]

Step 2: Convert Negative Exponents to Positive Exponents

The rule for negative exponents is that \[ a^{-n} = \frac{1}{a^n} \] Given our result from Step 1, \[ 9^{-5} \] we apply the rule to get:\[ \frac{1}{9^5} \]

Step 3: Evaluate the Expression

Now we calculate \(9^5\):\[ 9^5 = 9 \times 9 \times 9 \times 9 \times 9 \]Calculating step by step:- \(9 \times 9 = 81\)- \(81 \times 9 = 729\)- \(729 \times 9 = 6561\)- \(6561 \times 9 = 59049\)So, \(9^5 = 59049\) and the expression is:\[ \frac{1}{59049} \]

Key Concepts

Negative Exponents

Understanding negative exponents is important in simplifying and manipulating mathematical expressions. When an exponent is negative, it means that the base is on the denominator side of a fraction. For instance, the expression \(a^{-n}\) is equal to \(\frac{1}{a^n}\).
This means you 'flip' the base to make the exponent positive. Instead of multiplying \(a\) by itself \(n\) times, you are dividing 1 by \(n\) multiplications of \(a\).
This concept is particularly useful in making expressions with exponents always positive, which is often required in final answers. Remember:

• A negative exponent indicates a reciprocal.
• Converting negative exponents helps in equilibrium of expressions.

Multiplication of Exponents

Exponents can be simplified through specific exponent rules. One of the primary rules is the multiplication of exponents, which states that if two exponents have the same base, you can add the exponents. This rule is known as:
\[ a^m \times a^n = a^{m+n} \]
Hence, when you multiply \(9^{-4}\) by \(9^{-1}\), it involves adding the exponents \(-4\) and \(-1\) to get \(-5\).
This rule ensures calculations are quick and straightforward, avoiding the need to perform several multiplications. Understanding this rule helps tremendously in simplifying expressions that involve powers of the same base.

• The base remains the same.
• Add the exponents together.

Positive Real Numbers

Working with positive real numbers is essential when applying exponent rules. Real numbers include all possible numbers along the number line, from negative to positive infinity, including fractions and irrational numbers.

• Every variable in an exponent operation is assumed to be a positive real number.
• This assumption helps simplify the handling of expressions, ensuring that all operations involving exponents are well-defined.

For example, within the context of exponents, assuming the numbers involved are positive helps ensure the expressions remain manageable and that the operations, especially involving negative exponents, yield real-number results.
This makes it easier to apply exponent rules without worrying about undefined behaviors or imaginary numbers.