Problem 6 - Algebra 1: Concepts and Skills

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 form of the given expression is \(\frac{1}{5}\).

Step 1: Identify the base and the powers

In this problem, both the numerator and denominator share a common base which is 5. The powers in the numerator of the fraction are 3 and 5, and there is a power of 9 operates in the denominator.

Step 2: Apply the properties of exponents in the numerator

If you are multiplying with the same base, in this case being 5, we add the exponents. This stems from the product of powers property, \(a^m * a^n = a^{m+n}\). So \(5^{3} \cdot 5^{5} = 5^{3+5}\) which equals \(5^8\). So the expression becomes \(\frac{5^8}{5^9}\)

Step 3: Apply the quotient of powers property

Now, we apply the quotient of powers property, which states that when dividing with the same base, we subtract the exponent in the denominator from the exponent in the numerator. So the answer is \(5^{8-9} = 5^{-1}\)

Step 4: Convert from negative exponent to positive exponent

When the exponent is negative, to rewrite the expression with a positive exponent, we have to take the reciprocal of the base. So the answer \(5^{-1}\) can be written as \(\frac{1}{5}\).