91Ó°ÊÓ

When simplifying expressions involving exponents, it’s important to understand how the power of a power rule works. This rule tells us how to handle an exponent raised to another exponent. Imagine you have something like \( (a^m)^n \). This means you raise \(a\) (the base) to the power of \(m\), and then take that result and raise it to the power of \(n\).

According to the power of a power rule, \( (a^m)^n = a^{m \times n} \). This means you multiply the exponents together.

Let's use the example from the original exercise to see this in action: \( \left(z^{3}\right)^{5}\). You would multiply the exponents \( (3 \times 5)\) to get \( z^{15} \).

Another term in the expression is \( \left(z^{2}\right)^{3} \). Using the power of a power rule, multiply the exponents \( (2 \times 3)\) to get \( z^{6} \).

The power of a power rule is very handy for simplifying complex exponent expressions. Remember always to multiply the exponents when you see one power raised to another.

The product of powers rule helps us combine terms that have the same base but different exponents. When you multiply bases that are the same, you simply add the exponents.

The rule can be expressed as \( a^m \times a^n = a^{m+n} \). This means you keep the base the same and add the exponents together.

Let's go back to our simplified terms from the previous section: \(z^{15}\times z^{6} \). According to the product of powers rule, you add the exponents \ (15 + 6)\ to get \ (z^{21})\ as the final, simplified expression.

This method ensures you can easily combine multiple terms with the same base, making calculations quicker and simpler.

Simplifying expressions involves using rules like the power of a power rule and the product of powers rule to turn a complex expression into a simpler form. Effective simplification makes it easier to understand and work with the expression.

Let's consider the original expression: \( \left(z^{3}\right)^{5}\left(z^{2}\right)^{3}\).

First, apply the power of a power rule:
- For \( \left(z^{3}\right)^{5} \), you get \(z^{15} \) (since \ (3\times 5 =15 \)).
- For \( \left(z^{2}\right)^{3} \), you get \(z^{6} \) (since \( 2 \times 3 =6 \)).

Next, use the product of powers rule:
- Combine the terms: \(z^{15}\times z^{6} \).
- Add the exponents: \( 15+ 6 \) to get \(z^{21} \.

In summary, simplifying expressions requires you to apply these exponent rules systematically. This will turn complicated expressions into much more manageable ones.