Problem 108 - Algebra and Trigonometry

Question

Simplify each exponential expression. Assume that variables represent nonzero real numbers. $$ \frac{\left(x y^{-2}\right)^{-2}}{\left(x^{-2} y\right)^{-3}} $$

Step-by-Step Solution

Answer

\(\frac{\left(x y^{-2}\right)^{-2}}{\left(x^{-2} y\right)^{-3}} = x^{8}y^{7}\

Step 1: Distribute the negative exponents

First, apply the power of a product property which states that \((ab)^n = a^n b^n\). This gives us \((x^{-2}y^4)/(x^6 y^{-3})\).

Step 2: Simplify the expression

Next, further simplify the expression by subtracting the exponents of like terms in the numerator and denominator. We get \(x^{6 - (-2)} y^{4 - (-3)}\).

Step 3: Evaluate the exponents

Finally by evaluating the exponents, we get \(x^{8}y^{7}\).

Step 4: Final Solution

So, the simplified form of the expression: \(\frac{\left(x y^{-2}\right)^{-2}}{\left(x^{-2} y\right)^{-3}}\) is \(x^{8}y^{7}\).

Key Concepts

Understanding Negative Exponents

When working with exponential expressions, it's essential to understand how negative exponents function. A negative exponent, such as in the expression \( b^{-n} \), indicates that the base (\( b \)) is on the wrong side of a fraction line. In simpler terms, a negative exponent tells you to take the reciprocal of the base raised to the positive of that exponent.
For example:

• \( y^{-2} \) becomes \( \frac{1}{y^2} \)
• \( x^{-3} \) becomes \( \frac{1}{x^3} \)

Negative exponents can be tricky, but remember, they simply mean "invert the base and make the exponent positive." This understanding will help you simplify expressions by converting negative exponents to positive ones.

The Power of a Product Property

The power of a product property is a fundamental rule when dealing with exponents. It states that when you have an expression like \( (ab)^n \), you can distribute the exponent to each factor inside the parenthesis.
This means:

• \( (xy)^{-2} \) becomes \( x^{-2}y^{-2} \)
• \( (x^{-2}y)^{-3} \) becomes \( x^{6}y^{-3} \)

In the given exercise, applying this property is key to simplifying complex expressions. First, distribute the negative exponent to each term in the product, turning the entire expression into something easier to handle by transforming each factor individually.

Simplifying Exponents in Fractions

Once you have distributed the exponents, the next step is to simplify the expression. This involves combining the exponents of like bases by performing operations between the exponents in the numerator and the denominator.
For instance, in the expression: \( \frac{x^{-2}y^{4}}{x^{6}y^{-3}} \)
Follow these steps:

• Combine the exponents of the base \( x \): \( x^{-2-6} \rightarrow x^{6-(-2)} = x^{8} \)
• Combine the exponents of the base \( y \): \( y^{4-(-3)} = y^{7} \)

Remember, when dividing like bases, you subtract the exponent in the denominator from the exponent in the numerator. Mastering this technique is essential for simplifying any expression involving exponents.

Understanding Variable Expressions

Variable expressions often include variables like \( x \) and \( y \), which can take any real number value.
In exponential expressions, these variables are commonly raised to various powers. Simplifying a variable expression involves understanding the behavior of these powers and applying the rules of exponents correctly.

• Assume all variables are nonzero: In most exercises, variables like \( x \) and \( y \) are nonzero to avoid undefined operations such as division by zero.
• Pay attention to like terms: This helps in combining or canceling terms effectively.

By grasping the rules and characteristics of variable expressions and exponents, solving such expressions becomes much more straightforward, paving the way for more advanced algebraic manipulations.