Problem 56 - Algebra and Trigonometry

Question

\(47-72\) . Simplify the expression, and eliminate any negative exponent(s). $$ \left(3 x^{-1} y^{-2}\right)^{-2}\left(x^{2} y^{5}\right)^{-1} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{1}{9y} \).

Step 1: Simplify the First Expression

The expression \( \left(3 x^{-1} y^{-2}\right)^{-2} \) should be simplified. Apply the power of a product rule: \((ab)^n = a^n b^n\).Simplify each part:- \(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\)- \((x^{-1})^{-2} = x^{2}\)- \((y^{-2})^{-2} = y^{4}\)Thus, \( \left(3 x^{-1} y^{-2}\right)^{-2} = \frac{1}{9} x^2 y^4 \).

Step 2: Simplify the Second Expression

For the expression \( (x^2 y^5)^{-1} \), apply the negative exponent rule: \(a^{-n} = \frac{1}{a^n}\).Simplify:- \((x^2)^{-1} = x^{-2}\)- \((y^5)^{-1} = y^{-5}\)Thus, \( (x^2 y^5)^{-1} = x^{-2} y^{-5} \).

Step 3: Combine the Simplified Expressions

Multiply the results from Step 1 and Step 2:\[\left( \frac{1}{9} x^2 y^4 \right) \times (x^{-2} y^{-5})\]Use the rules for multiplying exponents: \(a^m \times a^n = a^{m+n}\).Multiply:- Coefficient: \( \frac{1}{9} \)- For \(x\): \(x^{2 + (-2)} = x^0 = 1\)- For \(y\): \(y^{4 + (-5)} = y^{-1}\)Therefore the expression simplifies to \( \frac{1}{9} \cdot 1 \cdot y^{-1} = \frac{y^{-1}}{9} \).

Step 4: Eliminate Negative Exponent

Convert \( y^{-1} \) to eliminate the negative exponent using the rule: \(a^{-n} = \frac{1}{a^n}\).This becomes:\[\frac{1}{9} \cdot \frac{1}{y} = \frac{1}{9y}\]

Key Concepts

Power of a Product Rule

Understanding how to simplify expressions involving exponents starts with mastering the power of a product rule. In essence, this rule helps you distribute an exponent over a product of terms. If you have

• \((ab)^n = a^n b^n\)

this formula tells us that the exponent applies individually to each factor in the product, not just to the product as a whole.
For example, when dealing with the expression \((3 x^{-1} y^{-2})^{-2}\), you apply the exponent

• \((3)^{-2}\): which simplifies as \(\frac{1}{9}\),
• \((x^{-1})^{-2}\): modifies to \(x^2\),
• \((y^{-2})^{-2}\): simplifies into \(y^4\).

By understanding this rule, you can easily simplify the expression by calculating each component separately before recombining them.

Simplifying Algebraic Expressions

Simplifying a complex algebraic expression involves breaking it down into simpler parts. You modify each part according to algebraic rules, making them easier to work with. For an expression like \((x^2 y^5)^{-1}\), you use:

• Negative exponent rule (\(a^{-n} = \frac{1}{a^n}\))

Applying this rule:

• \((x^2)^{-1} = x^{-2}\)
• \((y^5)^{-1} = y^{-5}\)

Through these simplifications, the expression is transformed.
Once you have a clearer form, as in this exercise, you are ready to combine the results.
Successful simplification often depends on your ability to correctly apply these rules to each algebraic expression component.

Exponent Rules

Exponent rules provide an essential framework for working with expressions involving powers. Here are some key rules that are especially useful:

• Product of Powers Rule: \(a^m \times a^n = a^{m+n}\)
• Quotient of Powers Rule: \(a^m \div a^n = a^{m-n}\)
• Power of a Power Rule: \((a^m)^n = a^{m \times n}\)

In our exercise, after breaking down the original expression using these rules, you multiply:
\(\left( \frac{1}{9} x^2 y^4 \right)\) with \(x^{-2} y^{-5}\). Applying the product of powers rule simplifies to:

• \(x^{2 + (-2)} = x^0 = 1\)
• \(y^{4 + (-5)} = y^{-1}\)

Thus resulting in \(\frac{y^{-1}}{9}\). Eliminate negative exponents finally yields \(\frac{1}{9y}\).
Mastering these rules, the simplification becomes straightforward, enabling you to handle even challenging expressions with ease.