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The power rule is a key principle in algebra, especially when dealing with exponents. When you have a fraction raised to a power, like \(\left(\frac{a}{b}\right)^n\), the power rule tells us to apply the exponent to both the numerator \(a\) and the denominator \(b\). This means we'll get \(\frac{a^n}{b^n}\).
In our example, we have \(\left(\frac{4}{y}\right)^3\). By applying the power rule:

• The numerator 4 is raised to the power of 3.
• The denominator \(y\) is also raised to the power of 3.

After applying the power rule, we arrive at \(\frac{4^3}{y^3}\), which sets up the next steps in simplifying this expression.

Simplifying expressions in algebra involves reducing them to their simplest form. This process often involves performing operations like multiplication or division.When simplifying expressions with exponents, as in \(4^3\), you multiply the base (4 in this case) by itself as many times as the exponent indicates. So, \(4^3\) becomes \(4 \times 4 \times 4\). Calculating this gives us 64. Similarly, for \(y^3\), since \(y\) is the base, you multiply it by itself twice more, yielding \(y \times y \times y\).
Simplifying is crucial as it transforms complex expressions into a form that is easier to understand and work with. After simplifying \(\frac{4^3}{y^3}\) by performing these multiplications, we achieve a more straightforward expression: \(\frac{64}{y^3}\).

Algebraic fractions are fractions that contain algebraic expressions in the numerator, the denominator, or both. In our scenario, \(\frac{4}{y}\) is an algebraic fraction as it features a variable \(y\) in the denominator.
When dealing with exponentiation in algebraic fractions, the power rule helps us to simplify these expressions uniformly across both the numerator and denominator. This makes complex operations, such as raising a fraction to a power, manageable. In this exercise, after applying the power rule, we needed to handle \(\frac{4^3}{y^3}\). This illustrates that operations involving algebraic fractions can extend simple arithmetic rules, allowing for greater flexibility and precision in algebra. Understanding these components is essential in mastering solutions that involve algebraic fractions.