91Ó°ÊÓ

Understanding the power of power rule is crucial when dealing with exponential expressions. Essentially, this rule states that when you raise a power to another power, you multiply the exponents. For example, if we have \( (x^a)^b \), we calculate it as \( x^{a*b} \).

In our exercise, \(\left(-3 x^{-1} y^{2}\right)^{2}\), we apply this rule separately to each term inside the parentheses. There are three terms to consider: -3, \( x^{-1} \) and \( y^{2} \). The constant -3 is raised to the second power (\( (-3)^2 \)), the \( x^{-1} \) term is squared (\( (x^{-1})^2 \) becomes \( x^{-1*2} \) or \( x^{-2} \) ), and the \( y^2 \) term is also squared (\( (y^2)^2 \) becomes \( y^{2*2} \) or \( y^4 \) ). Simplifying using the power of power rule, our expression becomes \( 9*x^{-2}*y^4 \).

The negative exponent rule is an essential piece of simplifying expressions and can be initially counterintuitive. It tells us that a negative exponent indicates the reciprocal of the base raised to the opposite positive exponent. So, \( x^{-n} = \frac{1}{x^n} \).

In the provided example, after we have applied the power of power rule, we deal with a negative exponent in the term \( x^{-2} \). By applying the negative exponent rule, we move \( x \) to the denominator and change the exponent to positive, transforming the expression \( x^{-2} \) into \( \frac{1}{x^2} \). Thus, our partially simplified expression \( 9*x^{-2}*y^4 \) becomes a fraction, \( \frac{9y^4}{x^2} \).

Algebraic fractions are fractions that contain algebraic expressions in the numerator, the denominator, or both. Simplifying algebraic fractions can involve factoring, canceling common factors, and applying exponent rules.

When we arrive at the final step of our exercise, we find ourselves with the algebraic fraction \( \frac{9y^4}{x^2} \). This expression is already simplified as there are no common factors to cancel between the numerator and denominator. Algebraic fractions cannot always be interpreted as easily as numerical fractions, but understanding how to manipulate expressions with algebraic fractions by applying exponent rules, among others, adds to our toolkit for solving more complex algebra problems.