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The concept of the power of a power rule is essential when dealing with exponents raised by another exponent. This rule states that when you have an exponent raised to another exponent, you multiply the exponents. For example, in the expression \(\big(y^3\big)^9\), the outer exponent is 9, and the inner exponent is 3. According to the power of a power rule, you multiply these exponents together: \[ (y^3)^9 = y^{3 \times 9} \]. This simplifies to \[ y^{27} \].

This rule helps simplify complex expressions and makes calculations easier. It is one of the fundamental rules in algebra and prealgebra.

Multiplying exponents is directly related to the power of a power rule. When exponents are multiplied, especially in context with the power of a power, the process involves taking the exponents and multiplying them together to achieve a single simplified exponent.

Consider the expression \[ (y^3)^9 \]. According to the steps:

• Step 1: Identify the exponents. Here, the inner exponent is 3, and the outer exponent is 9.
• Step 2: Apply the power of a power rule by multiplying the two exponents: \[ 3 \times 9 \].
• Step 3: Perform the multiplication: \[ 3 \times 9 = 27 \].

So the expression simplifies to \[ y^{27} \].

This approach allows complicated exponent expressions to be easily simplified.

Prealgebra is the foundation of all higher-level mathematics, including algebra, geometry, and calculus. Understanding the basic rules of exponents is a crucial part of prealgebra. It includes learning simple rules such as the power of a power rule and multiplying exponents, which are fundamental for solving more complex equations and expressions later on.

In prealgebra, learning how to handle expressions like \( (y^3)^9 \), step by step, ensures students grasp the basics before tackling more advanced problems. Breaking down the steps clearly: identifying exponents, applying rules, and performing operations helps build a strong mathematical foundation.

With these well-understood, students can confidently move forward in their math journey.