Problem 14 - College Algebra

Question

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

Step-by-Step Solution

Answer

The simplified expression is \( \frac{1}{625} \).

Step 1: Apply the Division Rule for Exponents

To divide expressions with the same base, subtract the exponents. Here, the base is 5. So, \( 5^{-2} \div 5^{2} = 5^{-2 - 2} \).

Step 2: Simplify the Exponent

Simplify the exponent by performing the subtraction: \(-2 - 2 = -4\). Thus, you get \( 5^{-4} \).

Step 3: Convert to Positive Exponent

To express the answer with positive exponents, take the reciprocal of the base raised to the opposite sign of the exponent: \( 5^{-4} = \frac{1}{5^4} \).

Step 4: Calculate the Positive Power

Calculate \( 5^4 \) which is \( 5 \times 5 \times 5 \times 5 = 625 \). Thus, \( \frac{1}{5^4} = \frac{1}{625} \).

Key Concepts

Division Rule for Exponents

When simplifying expressions that involve exponents, one useful tool is the division rule for exponents. This rule applies when you are dividing two powers that have the same base, such as in the expression \( \frac{5^{-2}}{5^{2}} \). The division rule tells us to subtract the exponent in the denominator from the exponent in the numerator:

• The base remains the same (in this case, 5).
• The calculation simply involves subtracting the exponents \(-2 - 2 = -4\).

Once applied, the expression simplifies to \(5^{-4}\). It’s important to remember that this rule only works when the bases are alike, which makes it easy to simplify a complex expression by just handling the exponents directly.

Positive Exponents

Dealing with exponents often involves ensuring our final answers use positive exponents. Negative exponents might look intimidating, but they actually represent reciprocals of positive exponents. When you encounter a negative exponent, flip it to the denominator and change the sign:

• For example, convert \(5^{-4}\) into \(\frac{1}{5^4}\).
• The base (5) now remains in the denominator, raised to a positive power.

This operation transforms complex-looking terms into a more interpretable form, which is a standard practice in mathematics to ensure clarity and simplicity of expressions.

Exponent Rules

Exponent rules provide a framework for managing powers effectively. They are core to simplifying expressions where exponents appear. Here are some key rules to keep in mind:The Product Rule multiplies like bases by adding exponents:

• \(a^m \times a^n = a^{m+n}\)

The Quotient Rule divides like bases by subtracting exponents:

• \(\frac{a^m}{a^n} = a^{m-n}\)

The Power Rule elevates a power to a power by multiplying exponents:

• \((a^m)^n = a^{mn}\)

Lastly, the Zero Exponent Rule states that any non-zero base raised to the zero power is 1:

• \(a^0 = 1, \text{ where } a eq 0\)

These rules together create a cohesive set that helps to simplify and understand expressions with exponents easily. Keeping these in your repertoire makes dealing with complex expressions straightforward and manageable.