Problem 9 - Algebra 1

Question

Use the quotient of powers property to simplify the expression. $$ \frac{5^{3} \cdot 5^{5}}{5^{9}} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the base and the exponents

Identify the base that are the same in both the numerator and the denominator of the fraction. In this case, the base is 5 and the exponents in the numerator are 3 and 5 while the exponent in the denominator is 9.

Step 2: Apply the product of powers rule in the numerator

Apply the product of powers rule, which states that when you multiply two expressions with the same base you should add the exponents. So \(5^{3} \cdot 5^{5}\) becomes \(5^{3+5}\) this simplifies to \(5^{8}\). You now have \(\frac{5^{8}}{5^{9}}\).

Step 3: Apply the quotient of powers rule

Apply the quotient of powers rule. It says that when you divide two expressions with the same base, you should subtract the exponents. So, \(\frac{5^{8}}{5^{9}}\) simplifies to \(5^{8-9}\) which simplifies further to \(5^{-1}\).

Step 4: Simplify the negative exponent

We have a negative exponent, so we know that this number is a fraction. A negative exponent can be rewritten as a fraction under 1 by making the exponent positive, which gives us \(\frac{1}{5^{1}}\) or just \(\frac{1}{5}\).

Key Concepts

Product of Powers Rule

Understanding the product of powers rule is essential when working with exponents. It's a straightforward concept: if you have two exponential expressions with the same base being multiplied, simply add the exponents together to combine them into a single expression.

For example, when you see an expression like \(5^{3} \times 5^{5}\), both parts have the same base, 5. To simplify this, we add the exponents, which gives us \(5^{3+5}\) or \(5^{8}\). This property can greatly simplify calculations and help you to manage more complex expressions in algebra.

Simplifying Expressions with Exponents

Simplifying expressions with exponents can transform a seemingly complex mathematical problem into an easier one. Once you've applied rules like the product of powers, the next step often involves quotient of powers or dealing with negative exponents.

As we continue with the example, after applying the product of powers rule, we are left with \(\frac{5^{8}}{5^{9}}\). Here, we have the quotient of powers property in play. It tells us that for two exponents with the same base divided by each other, subtract the exponent in the denominator from the exponent in the numerator, resulting in \(5^{8-9} = 5^{-1}\). This process is paramount in simplifying expressions, making them more comprehensible and operable.

Negative Exponent

A negative exponent may at first be perplexing, but it follows a simple principle. An expression with a negative exponent like \(5^{-1}\) is equivalent to the reciprocal of that base with a positive exponent. Put differently, for \(a^{-n}\), you can rewrite it as \(\frac{1}{a^{n}}\), which often clarifies the value.

In our preceding example, \(5^{-1}\) simplifies to \(\frac{1}{5}\text{ or }5^{-1}\). This is because a negative exponent indicates the number of times 1 is divided by the base with a positive exponent. Recognizing how to handle negative exponents is a fundamental part of simplifying expressions, allowing for more efficient problem-solving in algebra.