91Ó°ÊÓ

In mathematics, exponentiation is a fundamental operation, and understanding its rules can make many problems easier. The power of a power rule is one of these essential principles that comes in handy when dealing with nested exponents. When you see an expression like \((a^m)^n\), it can be simplified using this rule. The rule states that you can multiply the exponents together: \(a^{m \cdot n}\).
Let's put this into practice with \((4^4)^{3/2}\). Here, the base \(a = 4\), the first exponent \(m = 4\), and the second exponent \(n = \frac{3}{2}\). Applying the power of a power rule, we calculate \(4^{4 \times \frac{3}{2}} = 4^6\).
By reducing it to a single exponent, expressions become more manageable and easier to compute, especially for larger numbers.

Simplifying expressions is a critical skill, especially in algebra. With exponents, simplifying often involves applying rules like the power of a power rule or combining like terms.
For the expression \((4^4)^{3/2}\), simplifying it through the power of a power rule reduces it to a straightforward single exponent of \(4^6\). As a result, we transform a potentially complex expression into one that is much simpler to work with.
Simplifying expressions not only makes calculations more manageable but also reduces the chance of errors in later steps. We can now focus on evaluating \(4^6\) without worrying about handling fractional powers during the calculation.

When evaluating powers like \(4^6\), breaking it down into smaller, more manageable steps can make the computation easier. Evaluating a power means finding out what number you get when you multiply a base by itself a certain number of times.
Let's evaluate \(4^6\):

• Start with finding \(4^3\). Compute \(4 \times 4 \times 4 = 64\).
• Square the result to find \(4^6\). So, \(4^6 = (4^3)^2 = 64^2\).
• Finally, calculate \(64 \times 64\) to get \(4096\).

Breaking it down this way allows you to handle even large exponents without feeling overwhelmed. Thus, the final result of \(4^6\) is \(4096\), which is the value of the given expression \((4^4)^{3/2}\).