Problem 26 - 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. $$5^{-2} \cdot 5^{-6}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{5^8}\).

Step 1: Identify the property of exponents used

To simplify the expression \(5^{-2} \cdot 5^{-6}\), we'll use the property of exponents that states \(a^m \cdot a^n = a^{m+n}\). This property allows us to add the exponents when the bases are the same.

Step 2: Apply the property of exponents

Since both terms in the expression have the same base (5), we add their exponents as follows:\(5^{-2} \cdot 5^{-6} = 5^{-2 + (-6)} = 5^{-8}\).

Step 3: Convert to positive exponents

The problem specifies that the final answer must have positive exponents. The reciprocal of a number with a negative exponent can be written with a positive exponent. So, \(5^{-8}\) can be rewritten as \(\frac{1}{5^8}\).

Key Concepts

Negative Exponents

Negative exponents can sometimes seem tricky, but they are easier to understand when we remember one simple rule. A negative exponent means the reciprocal of the positive exponent. For instance, if you see something like \(5^{-3}\), you can rewrite it as \(\frac{1}{5^3}\).
Here's why this works:

• When you have a negative exponent, it represents dividing by that number multiple times.
• The reciprocal is essentially the "flipping" of the number, turning a multiplication problem into a division problem.
• So, with \(5^{-3}\), you multiply the reciprocal, \(\frac{1}{5^3}\), instead of multiplying by \(5\) directly.

Understanding negative exponents is as simple as flipping the base to the other side of the fraction line and making the exponent positive.

Positive Exponents

Positive exponents are the classic exponents most learners already know. When you see a positive exponent, it indicates repeated multiplication of the base number. For example, in \(3^4\), you multiply 3 by itself four times.
Let's explore why we use positive exponents:

• Positive exponents tell us how many times to use the base in a multiplication.
• They are straightforward and show growth or increase, like scaling up numbers.
• With positive exponents, the base number becomes larger with each multiplication.

In simpler terms, when you see \(4^2\), think of it as \(4 \times 4\). Positive exponents are about making the number bigger through repeated multiplication.

Properties of Exponents

The properties of exponents are like the grammar rules of math involving exponents. These rules help simplify calculations and solve problems efficiently. Here are the main properties:

• **Multiplying with the Same Base:** If you have the same base with different exponents, \(a^m \cdot a^n = a^{m+n}\). This means you add the exponents together when multiplying a common base.
• **Power Raised to a Power:** If you have an exponent raised to another exponent, \((a^m)^n = a^{m \cdot n}\). This means you multiply the exponents.
• **Division with the Same Base:** When dividing with the same base, \(\frac{a^m}{a^n} = a^{m-n}\). This means you subtract the exponents.

Taking our original exercise as an example, we had \(5^{-2} \cdot 5^{-6}\). Using the property \(a^m \cdot a^n = a^{m+n}\), we add the exponents to get \(5^{-8}\). Properties like these turn complex problems into manageable steps.