Problem 28 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the given expression. Write answers with positive exponents. \(\left(\frac{x^{-3}}{y^{2}}\right)^{-5}\)

Step-by-Step Solution

Answer

The simplified expression is \(x^{15}y^{10}\).

Step 1: Apply the Power of a Quotient Property

The expression we have is \(\left(\frac{x^{-3}}{y^2}\right)^{-5}\). Start by applying the power of a quotient property: \(\left(\frac{a}{b}\right)^{m} = \frac{a^m}{b^m}\). This gives us \(\frac{x^{-3(-5)}}{y^{2(-5)}}\).

Step 2: Simplify the Exponents

Now simplify the exponents from the previous step. The numerator becomes \(x^{15}\) and the denominator becomes \(y^{-10}\). So the expression is \(\frac{x^{15}}{y^{-10}}\).

Step 3: Rewrite with Positive Exponents

Expressions with negative exponents can be rewritten with positive exponents by taking the reciprocal. For \(y^{-10}\), move it to the numerator to make it \(y^{10}\). Hence our expression is \(x^{15}y^{10}\).

Key Concepts

Power of a Quotient Property

When we deal with expressions where both a numerator and denominator are raised to an exponent, we can use the Power of a Quotient Property. This property helps us simplify such expressions effectively.
This property states that when you have a fraction raised to a power, like \( \left(\frac{a}{b}\right)^{m} \), it's equivalent to raising both the numerator and the denominator to that power, resulting in \( \frac{a^m}{b^m} \).
Here’s a simple way to think about it:

• The exponent \( m \) applies to both the top \( a \) and bottom \( b \) of the fraction.
• You express it as \( a \) raised to \( m \) over \( b \) raised to \( m \), separately.

Using this property makes simplifying more manageable, as you work with each part of the fraction independently.

Negative Exponents

Negative exponents might look a bit tricky, but they’re actually quite easy to handle once you understand what they mean.
A negative exponent tells you to take the reciprocal of the base. So, \( a^{-n} = \frac{1}{a^n} \).
Let’s break this down:

• If you see a negative exponent, think "flip it," meaning you move the base to the other part of the fraction.
• For example, \( x^{-3} \) becomes \( \frac{1}{x^3} \) if currently in the numerator.
• If \( x^{-3} \) is in the denominator, it flips to the numerator as \( x^3 \).

This rule helps convert expressions into a form that is easier to work with, especially connecting to the requirement to express answers using positive exponents.

Simplifying Expressions

Simplifying expressions is about making them as compact and clear as possible, while keeping the expression equal to its original form. When simplifying, we often need to rewrite exponents and combine like terms. Here’s a typical strategy:

• Apply any relevant properties, like the power of a quotient property, to rewrite the expression.
• Utilize negative exponents by rewriting them using their positive counterparts.
• Bring your expression to its most reduced form by handling both numerical coefficients and like terms separately.

For instance, with \( \frac{x^{15}}{y^{-10}} \), pushing \( y^{-10} \) into the numerator gives us \( x^{15}y^{10} \).
This way, we manage to express everything with positive exponents and in a simplified, neat format.