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The power of a power rule is a handy tool in algebra that helps simplify expressions involving exponents. It applies when you have an exponent raised to another exponent, such as \( (2a^3)^4 \).

The rule states that you multiply the exponents together. This means \( a^{m \times n} \) if you have \( (a^m)^n \). In our original exercise, \( (2a^3)^4 \) becomes \( 2^4 \times (a^3)^4 \). Simplifying this further, we calculate \( 2^4 = 16 \) and multiply exponents for \( a \: 3 \times 4 = 12 \).

Therefore, you achieve a simplified expression of \( 16a^{12} \). This rule is essential in breaking down expressions into more manageable chunks, making the solving process easier.

The multiplication rule for exponents typically appears when dealing with expressions having coefficients and variables. In our exercise, we have reached the expression \( 3b^2 \times 16a^{12} \).

This rule states that when you multiply numbers with the same base, you add the exponents. However, if the bases are different, as in this case, you simply multiply the coefficients—and keep the variables aside.

For example:

• Multiply the coefficients: \( 3 \times 16 = 48 \)
• Keep exponents separate for variables like \( a^{12} \) and \( b^2 \)

Thus, the simplified form is \( 48a^{12}b^2 \). This clear-cut approach avoids confusion and keeps calculations straightforward.

Simplifying expressions is a critical skill in algebra that involves reducing an equation to its most basic form. It can make complex problems more understandable.

Let's recap what we have done with our expression \( 3b^2 \times (2a^3)^4 \). Initially, we identified a portion where the power of a power rule applies, giving us \( 16a^{12} \).

We then used the multiplication rule for exponents to handle different bases, resulting in \( 48a^{12}b^2 \). So, the original expression simplifies to this tidy algebraic form.

Simplifying helps in solving equations efficiently and is used across various math fields, from simple algebra to higher-level calculus. It lays the groundwork for a deeper understanding of how numbers and variables interact. Understanding these foundational steps is the key to mastering more complex problems.