Problem 33 - Algebra and Trigonometry

Question

\(29-46\) Simplify each expression. $$ w^{-2} w^{-4} w^{6} $$

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Understand the expression

The given expression is \( w^{-2} w^{-4} w^{6} \). This expression involves powers of the same base, \( w \). We will simplify this by using the properties of exponents.

Step 2: Apply the Product of Powers Property

According to the product of powers property, \( a^m \times a^n = a^{m+n} \). Using this property, we can combine the powers of \( w \): \( w^{-2} \times w^{-4} \times w^{6} = w^{-2 + (-4) + 6} \).

Step 3: Calculate the Exponent Sum

Add the exponents: \( -2 + (-4) + 6 = -2 - 4 + 6 \). Simplify this to find the sum: \( -2 - 4 = -6 \), then \( -6 + 6 = 0 \).

Step 4: Simplify the Expression with Zero Exponent

According to the zero exponent rule, any number (except zero) raised to the power of zero is 1, i.e., \( a^0 = 1 \). Thus, \( w^0 = 1 \).

Key Concepts

Product of Powers Property

When dealing with expressions that have multiple exponents and the same base, the product of powers property is incredibly useful. This rule tells us how to manipulate and simplify these expressions effectively.

The product of powers property states that when you multiply like bases, you can add their exponents. In mathematical terms, this is expressed as \( a^m \times a^n = a^{m+n} \). By employing this property, we can reduce the complexity of expressions like \( w^{-2} w^{-4} w^{6} \) by combining the powers of the base \( w \). To apply this rule to our example:

• Let's add the exponents: \( -2 + (-4) + 6 \).
• This turns the expression into \( w^{-2 + (-4) + 6} \).

Simplifying the exponent enables us to handle expressions with ease, cutting down the number of operations required to simplify them.

Exponent Rules

Exponent rules are a set of guidelines that dictate how to perform operations like multiplication and division of numbers with exponents and how to manage their powers. There are several key rules to understand when working with exponents:

• Product of Powers: As we discussed before, this rule helps in combining exponents with the same base, using additive calculation.
• Power of a Power: If you have an exponent raised to another exponent, you multiply the exponents together: \( (a^m)^n = a^{m \times n} \).
• Power of a Product: When applying an exponent to a product, it is applied to each factor: \( (ab)^n = a^n \times b^n \).
• Negative Exponent: A negative exponent signifies reciprocation: \( a^{-n} = \frac{1}{a^n} \).

Understanding these rules is essential for simplifying complex exponent expressions easily and accurately.

Zero Exponent Rule

The zero exponent rule is a fundamental exponent rule that simplifies expressions significantly. According to this rule, any non-zero number raised to the power of zero is 1: \( a^0 = 1 \). This concept is incredibly helpful when simplifying expressions.

In the context of our problem, after applying the product of powers property and combining all the exponents, we end up with the exponent zero. Thus, our expression \( w^{0} \) simplifies to 1.

It's important to remember that this rule does not apply to the base being zero, as \( 0^0 \) is considered indeterminate and does not follow this rule. Mastering the zero exponent rule allows for quick simplification of expressions where the sum of the exponents can lead to zero, resulting in a much simpler outcome.