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Chapter 10: Problem 264

In the following exercises, simplify. $$\left(\frac{x^{3} \cdot x^{8}}{x^{4}}\right)^{3}$$

Short Answer

Expert verified
The simplified expression is \(x^{21}\).

Step by step solution

01

Simplify the expression inside the parentheses

Combine the exponents in the numerator: \(x^{3} \times x^{8} = x^{3+8} = x^{11}\), so the expression becomes \(\frac{x^{11}}{x^{4}}\).
02

Simplify the fraction

Subtract the exponents in the fraction: \(\frac{x^{11}}{x^{4}} = x^{11-4} = x^{7}\). Now the expression inside the parentheses is \(x^{7}\).
03

Apply the outer exponent

Raise the simplified expression to the power of 3: \((x^{7})^{3} = x^{7 \times 3} = x^{21}\).

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Key Concepts

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

Exponent Rules
Understanding exponent rules can make simplifying expressions much easier. In our exercise, we use two main exponent rules:

1. **Product of Powers Rule**
This rule states that when you multiply two powers with the same base, you add the exponents. For example: \(x^a \times x^b = x^{a+b}\).

2. **Power of a Quotient Rule**
This rule tells us to subtract the exponents when dividing two powers with the same base. For instance: \(\frac{x^a}{x^b} = x^{a-b}\).

Using these rules makes it possible to combine and simplify expressions efficiently.
Simplifying Fractions
Simplifying fractions with exponents involves using the rules for dividing powers. In our exercise, we had to simplify the fraction \(\frac{x^{11}}{x^4}\). This was done by subtracting the exponents, giving us \(x^{11-4} = x^7\).

Here's a step-by-step guide for simplifying fractions with exponents:
  • Identify the base that is common in the numerator and the denominator.
  • Apply the power of a quotient rule: Subtract the exponent in the denominator from the exponent in the numerator.
  • Write the simplified expression.
By following these steps, fractions with exponents become much easier to handle.
Multiplying Exponents
When multiplying exponents, especially those raised to another power, there is another key rule to follow called the **Power of a Power Rule**. This rule states that when you have an exponent raised to another exponent, you multiply the exponents together. For example: \((x^a)^b = x^{a \times b}\).

In the final step of our exercise, we applied this rule to simplify \( (x^7)^3 \), resulting in \( x^{7 \times 3} = x^{21} \). This makes it easier to handle more complex expressions by breaking them down into our simpler rules.

So remember, when you see an exponent raised to another exponent, just multiply the two exponents to get your simplified result.

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Most popular questions from this chapter

Multiply the binomials using various methods. $$(5 u-3)(u+8)$$

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