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Chapter 10: Problem 353

In the following exercises, simplify. $$x^{4} \cdot x^{-2} \cdot x^{-3}$$

Short Answer

Expert verified
\( \frac{1}{x} \)

Step by step solution

01

- Apply Product of Powers Property

Use the product of powers property, which states that when multiplying exponential expressions with the same base, you add the exponents: \[ x^a \times x^b \times x^c = x^{a+b+c} \]
02

- Combine the Exponents

Combine the exponents for the terms \( x^4 \), \( x^{-2} \), and \( x^{-3} \) as follows: \[ x^{4 + (-2) + (-3)} \]
03

- Simplify the Exponents

Calculate the sum of the exponents: \[ x^{4 - 2 - 3} = x^{-1} \]
04

- Express in Standard Form

Express the result in standard form using the negative exponent rule, which states \( x^{-a} = \frac{1}{x^a} \). Therefore, \[ x^{-1} = \frac{1}{x} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Product of Powers Property
When dealing with exponents, the Product of Powers Property is very useful. It helps to simplify expressions where the same base is multiplied with different exponents. The rule states that if you have the same base with different exponents, you simply add the exponents together. For instance, if you have an expression like \(x^a \cdot x^b \cdot x^c\), you can combine these to form \(x^{a+b+c}\). This property makes it easy to simplify complicated exponential expressions into a single term. Always remember to first check if all the bases are the same. If they are, you can easily combine the exponents.
Combining Exponents
Next, let's combine the exponents. Using the expression from the exercise, \(x^4 \cdot x^{-2} \cdot x^{-3}\), we apply the Product of Powers Property. The exponents are 4, -2, and -3. According to the rule, we combine them as follows:
  • Add 4 with -2: \(4 + (-2) = 2\)
  • Add the result with -3: \(2 + (-3) = -1\)
By combining these exponents, we get the simplified term \(x^{-1}\). This part is crucial as it reduces the complexity of the expression and makes it more manageable. Always perform the addition carefully, paying close attention to positive and negative signs.
Negative Exponent Rule
Finally, let's handle the negative exponent. The Negative Exponent Rule states that any base with a negative exponent can be expressed as the reciprocal of the base with a positive exponent. In mathematical terms, \(x^{-a} = \frac{1}{x^a}\). Applying this to our term \(x^{-1}\), we get:
  • \(x^{-1}\) becomes \(\frac{1}{x}\)
This rule helps us convert a negative exponent into a more standard form, which is often easier to interpret and use in further calculations. So, our simplified term \(x^{-1}\) is now written as \(\frac{1}{x}\). This conversion is vital for ensuring that expressions are in their simplest and most understandable form.

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