Question

Write your answer as a power or as a product of powers. $$ (-y)^{3}(-y)^{4}(-y)^{5} $$

Short answer

\(-y^{12}\)

Step-by-step solution

Identify the common base and exponents

The common base is \(-y\) and the exponents are \(3, 4, 5\). The expression is \((-y)^{3}(-y)^{4}(-y)^{5}\).

Apply the rule for multiplying like bases

Apply the exponent rule that states to add the exponents when multiplying like bases. This results in \(-y^{(3 + 4 + 5)}\).

Simplify the expression

Add the exponents together to get \(-y^{12}\).

Key concepts

Multiplying Like Bases

Understanding how to multiply like bases is crucial when working with exponential expressions. When you multiply expressions with the same base, instead of multiplying the bases themselves, you simply add the exponents. This is a shortcut that makes calculations much easier.

For example, if you have the expression \(a^m \cdot a^n\), where \(a\) is the base and \(m\) and \(n\) are the exponents, the rule states that you should add the exponents together: \(a^m \cdot a^n = a^{m+n}\).

This rule applies regardless of whether the base is a number, a variable, or a negative quantity. Properly applying this rule leads to a great simplification of expressions, making further algebraic manipulations easier.

Negative Bases in Exponents

When dealing with negative bases in exponents, it's important to pay attention to the presence of parentheses, as they significantly affect the outcome. For a base enclosed in parentheses, like \( (-a)^n\), the negative sign is raised to the power along with the base. This results in different outcomes depending on whether the exponent is even or odd:

• If \(n\) is even, the result is positive since a negative number raised to an even power equals a positive number.
• If \(n\) is odd, the result is negative because a negative number raised to an odd power remains negative.

If there are multiple factors with the same negative base being multiplied together, as in the exercise \( (-y)^3(-y)^4(-y)^5\), we need to apply the multiplying like bases rule, adding the exponents together while considering the overall sign of the result based on the rules mentioned above.

Simplifying Exponential Expressions

Simplifying exponential expressions involves applying a variety of exponent rules to condense the expression into its most compact form. Alongside the rule for multiplying like bases, other rules such as the power of a power rule and the rule for dividing like bases, which involves subtracting exponents, play a role in the simplification process.

Starting with the expression involving multiplication of like bases, you combine the exponents. After that, other simplifications may be possible depending on the context of the problem. For instance, in our example \( (-y)^{3}(-y)^{4}(-y)^{5} \), we combine the exponents to get \( (-y)^{12}\). From here, one needs to remember that since 12 is an even number, the negative base raised to this power will result in a positive outcome, as negative numbers to even powers are positive. Hence the final simplified expression, taking into account the negative base and all exponent rules, is \(y^{12}\).

Remember that simplification is not just about making expressions shorter; it's about making them clearer and more understandable so that further operations can be handled with ease.