Question

Problems 154-162 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Simplify \(\left(\frac{x^{2} y^{-3}}{x^{4} y^{5}}\right)^{-2} .\) Assume \(x \neq 0\) and \(y \neq 0 .\) Express the answer so that all exponents are positive.

Short answer

The simplified expression is \( x^{4} y^{16}.\)

Step-by-step solution

Apply the power of a quotient rule

The power of a quotient rule states ewline ewline \(\bigg( \frac{a}{b} \bigg)^n = \frac{a^n}{b^n} \)ewline ewline Apply this rule to the given expression: ewline ewline \(\bigg( \frac{x^{2} y^{-3}}{x^{4} y^{5}} \bigg)^{-2} = \frac{\big(x^{2} y^{-3} \big)^{-2}}{\big(x^{4} y^{5} \big)^{-2}}\)

Apply the power of a product rule

The power of a product rule states ewline ewline \((a \text{ } b)^n = a^n \text{ } b^n \)ewline ewline Apply this rule to the numerator and the denominator: ewline ewline \(\frac{x^{2(-2)} y^{-3(-2)}}{x^{4(-2)} y^{5(-2)}} = \frac{x^{-4} y^{6}}{x^{-8} y^{-10}}\)

Simplify the exponents

Combine the terms with the same base by subtracting the exponents: ewline ewline \(\frac{x^{-4} y^{6}}{x^{-8} y^{-10}} = x^{-4-(-8)} y^{6-(-10)} = x^{4} y^{16}\)

Express the answer with positive exponents

Ensure that all the exponents are positive. The simplified expression is: ewline ewline \( x^{4} y^{16} \)

Key concepts

power of a quotient rule

When simplifying expressions with exponents, the power of a quotient rule is incredibly useful. This rule states that \(\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}\). What this means is, if you have a fraction raised to an exponent, you can apply that exponent to both the numerator and the denominator separately.

For the given exercise, we start with \(\left(\frac{x^{2} y^{-3}}{x^{4} y^{5}}\right)^{-2} \). Applying the power of a quotient rule gives us:

\(\left( \frac{x^{2} y^{-3}}{x^{4} y^{5}} \right)^{-2} = \frac{\left( x^{2} y^{-3} \right)^{-2}}{\left( x^{4} y^{5} \right)^{-2}}\).

This step makes the expression easier to manage by separating the negative exponent from the fraction.

power of a product rule

The power of a product rule allows you to distribute an exponent over multiple terms inside a product. The rule is given by \( (ab)^n = a^n b^n \). Simply put, if you have a product of variables raised to an exponent, you can apply the exponent to each variable individually.

In our exercise, after applying the power of a quotient rule, we have \( \frac{(x^{2} y^{-3})^{-2}}{(x^{4} y^{5})^{-2}} \). Using the power of a product rule, this becomes:

\( \frac{x^{2(-2)} y^{-3(-2)}}{x^{4(-2)} y^{5(-2)}} = \frac{x^{-4} y^{6}}{x^{-8} y^{-10}} \).

Notice how the exponents are distributed over each term inside the parentheses. This step simplifies handling complex products raised to a power.

positive exponents

In algebra, it's often helpful to express all exponents as positive for a cleaner and more interpretable answer. An exponent tells us how many times a number is multiplied by itself, and a positive exponent is the standard way to present results.

In the given exercise, we land at \( \frac{x^{-4} y^{6}}{x^{-8} y^{-10}} \). Simplifying this involves combining the terms with the same base and handling negative exponents carefully.

Use the rule that \( a^{-n} = \frac{1}{a^n} \) to convert any negative exponents to positive ones. In this step, we simplify the fraction by subtracting the exponents of like bases:

\( \frac{x^{-4} y^{6}}{x^{-8} y^{-10}} = x^{-4-(-8)} y^{6-(-10)} = x^{4} y^{16} \).

Now, the expression has positive exponents.

simplifying algebraic expressions

Simplifying algebraic expressions involves reducing them to their most basic form without changing their value. This often includes combining like terms, canceling common factors, and converting negative exponents to positive ones.

In the final steps of our exercise, we simplify \( \left( \frac{x^{2} y^{-3}}{x^{4} y^{5}} \right)^{-2} \) down to \( x^{4} y^{16} \). By carefully applying exponent rules like the power of a quotient, power of a product, and handling negative exponents, the complex expression becomes manageable.

Remember the core steps:

• Apply exponent rules systematically.
• Combine like terms to simplify.
• Convert negative exponents to positive if needed.

Mastery of these techniques will make algebraic expression simplification much easier.