91Ó°ÊÓ

To begin, let's delve into the power of a power rule. This rule is very straightforward and important when dealing with exponents. The power of a power rule states that when you raise an exponent to another exponent, you multiply the exponents together. In mathematical terms, it can be written as:
\[(a^{m})^{n} = a^{m \times n}\] Here, we are dealing with a base 'a' raised to an exponent 'm', which is then again raised to an exponent 'n'.
The result is the base 'a' raised to the product of 'm' and 'n'.
For example:
If we have \((a^{2})^{3}\), we would multiply the exponents 2 and 3, resulting in \((a^{2 \times 3}) = (a^{6})\).
Similarly, for \((a^{3})^{2}\), we would multiply the exponents 3 and 2, resulting in \((a^{3 \times 2}) = (a^{6})\).
Now you see how this rule simplifies expressions involving exponents!

Simplifying expressions involving exponents can seem tricky at first, but with practice, it becomes second nature. Let's use what we learned from the power of a power rule to simplify expressions.
For example, when faced with \((a^{2})^{3}\), you should immediately think of multiplying the exponents.
So we simplify it to \((a^{2 \times 3}) = (a^{6})\).
On the other hand, for the expression \((a^{3})^{2}\), we apply the same rule and simplify it to \((a^{3 \times 2}) = (a^{6})\).
By following these steps, you can easily break down complex exponential expressions into simpler forms. This is an essential skill in algebra and beyond, as it helps to solve equations and understand functions more clearly.

Verifying equations is all about ensuring that both sides of an equation are equal. In this case, we were asked to verify if \((a^{2})^{3} == (a^{3})^{2}\).
1. Start by simplifying both sides of the equation separately using the power of a power rule.
2. For the left side \((a^{2})^{3}\), simplify to \((a^{6})\).
3. For the right side \((a^{3})^{2}\), simplify to \((a^{6})\).
4. If both simplified expressions match, then the given equation is true.
In our example, both sides simplified to \((a^{6})\), so the equation \((a^{2})^{3} == (a^{3})^{2}\) is verified as true.