91Ó°ÊÓ

Understanding the power rule is crucial when working with exponential expressions. It is a fundamental concept in algebra that allows you to simplify terms raised to powers. Here's the rule: when you raise a power to another power, you multiply the exponents. In mathematical terms, this is written as \( (a^m)^n = a^{mn} \). Let's see this in action with the given exercise.

When working with \( (-5v^6)^3 \), we apply the power rule separately to \( -5 \) and \( v^6 \). For \( -5 \), it's raised to the third power, which simply results in \( -5^3 \). For \( v^6 \) being raised to the third power, we apply the power rule by multiplying the exponents: \( 6 \) and \( 3 \. Hence our expression for \( v \) becomes \( v^{6 \times 3} = v^{18} \).

Simplifying expressions is a key skill in algebra that makes equations easier to work with. The process often involves applying a variety of rules like the power, product, and distributive laws. With the power rule understood, we focus on reducing the expression to its simplest form.

For this exercise, after applying the power rule, we have \( -5^3 \) and \( v^{18} \. The term \( -5^3 \) simplifies to \( -125 \. No additional simplification is required for \( v^{18} \. Finally, we just combine these results to get \( -125v^{18} \. It's important to remember that simplifying an expression doesn't change its value; it merely presents the value more neatly or efficiently.

Negative exponents can often be a source of confusion, but they follow a simple rule. An expression with a negative exponent, such as \( a^{-n} \), is equivalent to \( 1/a^n \. In other words, you take the reciprocal of the base raised to the absolute value of the exponent.

However, in our current exercise, the exponent is positive after simplification. But it's helpful to know that if we had a negative exponent in the expression, we would handle it by reciprocating the base and making the exponent positive. In general practice, if you encounter a negative exponent, remember that you're dealing with the inverse of the base raised to the power of the exponent without the negative sign.