Problem 14 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the given expression. Write answers with positive exponents. \(5^{-2} \div 5^{2}\)

Step-by-Step Solution

Answer

\(\frac{1}{625}\)

Step 1: Apply the Law of Exponents

The law of exponents states that when dividing powers with the same base, you subtract the exponents. Apply this rule to the given expression: \[ 5^{-2} \div 5^{2} = 5^{-2 - 2} = 5^{-4} \]

Step 2: Convert Negative Exponent to Positive

To express the result with positive exponents, use the rule that \(a^{-n} = \frac{1}{a^n}\). Apply this to the expression:\[ 5^{-4} = \frac{1}{5^{4}} \]

Step 3: Simplify Further if Necessary

Calculate the expression \(5^{4}\) if asked to express the result in numerical form:\[ 5^{4} = 5 \times 5 \times 5 \times 5 = 625 \]So, the simplified expression is:\[ \frac{1}{625} \]

Key Concepts

Understanding Negative Exponents

Negative exponents can seem confusing at first, but they follow a simple rule: they represent the reciprocal of the base raised to the corresponding positive exponent. In other words, when you see something like \( a^{-n} \), you can think of it as \( \frac{1}{a^n} \). This conversion is crucial because it enables us to express any calculations with positive exponents, which are often easier to interpret.
Let's consider the number \( 5^{-2} \). Instead of thinking of it as a negative power, we translate it to a fraction: \( \frac{1}{5^2} \). This means you would take \( 1 \) and divide it by \( 5 \) multiplied by itself, or \( 25 \).
Negative exponents, therefore, are just a mechanism to work with reciprocals easily without always explicitly working with fractions. They're a powerful concept that simplifies many mathematical operations.

Simplifying Expressions with Exponents

Simplifying expressions, especially those involving exponents, is about reducing them to their most basic forms while following set mathematical rules. When dealing with exponents, one fundamental rule is that when you divide two exponents with the same base, you subtract the exponents: \( a^m \div a^n = a^{m-n} \).
For the expression \( 5^{-2} \div 5^{2} \), apply this rule: subtract the exponent \( 2 \) from \(-2\), resulting in \( 5^{-4} \). The simplification makes the expression less complicated and more manageable.

• Always ensure to express your final result with positive exponents.
• Simplifying helps in reducing errors and eases further calculations.

If further simplification is required, perform additional arithmetic, just as when converting negative exponents to positive through reciprocals.

Using Positive Exponents

Positive exponents are straightforward. They tell us how many times to multiply the base by itself. For instance, \( 5^4 \) means multiplying \( 5 \) four times: \( 5 \times 5 \times 5 \times 5 \), which calculates to \( 625 \).
When an expression originally includes negative exponents, conversion to positive exponents makes it easier to evaluate practical computations. In the case of \( 5^{-4} \), which we've already transformed into \( \frac{1}{5^4} \), we can calculate the power of \( 5 \), resulting in the value \( 1/625 \).

• Always check if you can further simplify expressions to bare integers if applicable.
• Using positive exponents is not only beneficial for simplification but often required for final answers, especially in exams and homework.

Ensuring you work with positive exponents prevents misunderstandings and aligns with common mathematical practices.