Problem 34 - College Algebra

Question

Simplify each exponential expression $$ \left(x^{-6}\right)^{4} $$

Step-by-Step Solution

Answer

The simplified form of \( \left(x^{-6}\right)^{4}\) is \( \frac{1}{x^{24}}\)

Step 1: Understand the expressions and the rule

Here we have the term \(x^{-6}\) raised to the power 4. According to the power of power rule in exponents, if we have \( (x^{m})^{n}\) , the rule is to multiply these exponents to give \(x ^{mn}\). Also, a negative exponent represents the reciprocal of the base raised to that positive exponent. So, \(x^{-m} = \frac{1}{x^{m}}\)

Step 2: Apply the power of power rule

Applying the power of power rule here gives us \( x^{(-6*4)} = x^{-24}\)

Step 3: Simplify using the rule for negative exponents

Using the rule for negative exponents, we simplify \(x^{-24}\) to \( \frac{1}{x^{24}}\)

Step 4: Final Solution

So the final simplified version of \( \left(x^{-6}\right)^{4}\) is \( \frac{1}{x^{24}}\)

Key Concepts

Power of Power Rule

When dealing with exponential expressions, understanding the power of power rule is crucial. This rule is applied when you have an exponent raised to another exponent. For example, consider \[ (x^{m})^{n} \]This can be simplified by multiplying the exponents, resulting in \[ x^{mn} \]Using this method significantly reduces complex expressions. It combines two exponential operations into one, making it easier to work with.

In our exercise, we had \( (x^{-6})^{4} \)Remember to multiply the exponents: \[ -6 \times 4 = -24 \]So the expression becomes \( x^{-24} \)through the power of power rule. By simplifying first, you avoid unnecessary complications later in your calculations.

Negative Exponents

Negative exponents can seem tricky, but they're easier to handle once you understand what they mean. A negative exponent indicates that instead of multiplying by the base, you divide by it. In simple terms, a negative exponent tells you to "flip" the base into its reciprocal.

• A base raised to a negative exponent, like \( x^{-m} \), becomes \( \frac{1}{x^m} \)
• Always convert negative exponents to positive by taking the reciprocal of the base.

Applying this to our expression \( x^{-24} \),we convert it to \( \frac{1}{x^{24}} \).It's like saying "take 1 divided by \( x^{24} \)". This transformation is crucial when simplifying expressions, allowing for easier manipulation and interpretation of the result.

Reciprocal

The concept of a reciprocal is foundational in understanding mathematical operations involving division and negative exponents. The reciprocal of a number is simply one divided by that number.

• For any number \( a \), its reciprocal is \( \frac{1}{a} \).
• When dealing with exponential expressions, the reciprocal helps to manage negative exponents and simplify calculations.

In our worked example, we encountered \( x^{-24} \).By understanding negative exponents as reciprocals, we know this is equivalent to \( \frac{1}{x^{24}} \).Recognizing this step often shifts complex problems into simpler forms, making it easier to solve equations or integrate into further mathematical operations.