91Ó°ÊÓ

Understanding the negative power rule is crucial for simplifying exponential expressions. The rule states that any number or expression raised to a negative exponent is equal to one divided by that same expression with a positive exponent. In mathematical terms, this is shown as \(a^{-n} = \frac{1}{a^n}\).
This concept is useful for changing negative exponents into positive ones, which are generally easier to work with when simplifying expressions. For instance, if you have an expression like \(\left(\frac{3x^{4}}{y}\right)^{-3}\), using the negative power rule helps by transforming it to \(\frac{1}{\left(\frac{3x^{4}}{y}\right)^{3}}\).
Recognizing this step is essential as it sets up the expression for further simplification through other rules.

The power of a quotient rule allows us to handle expressions where a fraction is raised to an exponent. According to this rule, exponents can be distributed separately to the numerator and the denominator of the fraction. It can be expressed as:

• \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)

In the context of the exercise given, after converting the negative exponent to a positive one, we used the power of a quotient rule on \(\frac{1}{\left(\frac{3x^{4}}{y}\right)^{3}}\).
It allows us to rewrite the expression as \(\frac{1}{\left(3^3 \, x^{4 \times 3} \, / \, y^3\right)}\). Notice how the exponent of 3 is applied to both the numerator terms \(3\) and \(x^4\), as well as the denominator term \(y\).
Grasping this rule helps determine how each part of the fraction is affected by the exponent and to simplify it effectively.

Simplifying fractions in exponential expressions is an important skill that combines understanding exponents with basic fraction reduction. Once the exponents have been distributed using the power of a quotient rule, simplification involves calculating the final values for the numerator and denominator. For example, in the expression \(\frac{27x^{12}}{y^3}\), we simplify further by arranging the terms clearly,

• From \(3^3 = 27\)
• And \(x^{4 \times 3} = x^{12}\)
• So, the denominator \(y^3\) stays as it is.

This expression is already at its simplest form. However, recognizing when to rearrange or keep terms the same is key.
Finally, to express the solution in a typical fraction form, move the denominator below the line, yielding the answer \(\frac{y^3}{27x^{12}}\).
Simplifying correctly ensures the expression is fully reduced and as accessible as possible for interpretation.