Problem 50 - Algebra for College Students

Question

Simplify each expression. Express final results without using zero or negative integers as exponents. \(\left(x^{5} y^{-1}\right)^{-3}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{y^3}{x^{15}} \).

Step 1: Apply Power to Power Rule

When we raise a power to a power, we multiply the exponents. So, for \( \left(x^5 y^{-1}\right)^{-3} \), we apply the power to both exponents inside the parentheses. This gives us \( x^{5 \cdot (-3)} y^{-1 \cdot (-3)} \).

Step 2: Simplify the Exponents

Multiply the exponents using the results from Step 1. For the \( x \) term, \(5 \cdot (-3) = -15\), so we have \( x^{-15} \). For the \( y \) term, \(-1 \cdot (-3) = 3\), so we have \( y^{3} \).

Step 3: Rewrite with Positive Exponents

To express the final result with only positive exponents, we move any terms with negative exponents from the numerator to the denominator. \( x^{-15} \) becomes \( \frac{1}{x^{15}} \), so the expression \( x^{-15}y^3 \) becomes \( \frac{y^3}{x^{15}} \).

Key Concepts

Power to Power Rule

The Power to Power Rule is a fundamental principle in exponentiation. When we apply this rule, we raise a power to another power. This means you multiply the exponents together. Let's break it down:

Imagine you have an expression like \((a^m)^n\). To simplify it using the Power to Power Rule, simply multiply the exponents: \(a^{m \cdot n}\).

• This rule is especially helpful when dealing with expressions raised to additional powers, simplifying calculations efficiently.
• In the context of the given problem, \(\left(x^5 y^{-1}\right)^{-3}\), we apply the rule to each variable separately. For \(x\), it's \(x^{5\cdot(-3)}\), and for \(y\), it's \(y^{-1\cdot(-3)}\).
• Remember, the order of operations requires parentheses to be handled first, applying the Power to Power rule in nested structures.

By mastering this rule, you'll simplify complex expressions with multiple exponents, streamlining the problem-solving process.

Negative Exponents

Negative exponents can seem tricky, but they have a straightforward interpretation.

A negative exponent like \(a^{-n}\) means that you take the reciprocal of the base raised to the corresponding positive exponent. For instance, \(a^{-n} = \frac{1}{a^n}\).

• In practical terms, negative exponents indicate a division rather than multiplication.
• In the given exercise, we saw the \(x^{-15}\) term. Here, it converts to \(\frac{1}{x^{15}}\) to ensure a positive exponent.
• Remember, a common mistake is to forget converting variables with negative exponents. Double-check all terms!

Understanding negative exponents allows you to manipulate and simplify expressions fully, making them more intuitive and manageable.

Positive Exponents

Positive exponents represent repeated multiplication.

For any base \(a\), \(a^n\) means \(a\) multiplied by itself \(n\) times. Positive exponents are the default situation in math, bringing simplicity and clarity.

• Having positive exponents is often preferred as it reflects the natural multiplication process.
• In the context of the problem, \(y^3\) remains as is, since it already had a positive exponent after simplification.
• When converting from negative to positive exponents, as we described, it's crucial to adjust the expression as needed for a clearer mathematical statement.

Utilizing positive exponents ensures your final expressions are easy to read and use, crucial for further calculations or applications.