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Chapter 10: Problem 25

Simplify.$$\left(a^{11} b^{8}\right)^{3}$$

Short Answer

Expert verified
\( a^{33} b^{24} \)

Step by step solution

01

Apply the Power of a Product Rule

The exercise requires simplifying the expression \( \left( a^{11} b^8 \right)^3 \). Using the power of a product rule, which states that \( (xy)^n = x^n y^n \), we can distribute the exponent of 3 to both parts within the parentheses.
02

Distribute the Exponent

Apply the exponent to each factor inside the parentheses. So, \( \left( a^{11} b^8 \right)^3 = (a^{11})^3 \times (b^8)^3 \).
03

Apply the Power of a Power Rule

Now, use the power of a power rule, which says \( (x^m)^n = x^{m \times n} \). Thus, simplify each factor separately: \((a^{11})^3 = a^{11 \times 3} = a^{33} \) and \((b^8)^3 = b^{8 \times 3} = b^{24} \).
04

Combine the Results

Combine the simplified expressions from the previous step: \( a^{33} \times b^{24} \).
05

Final Simplified Expression

The final simplified expression is \( a^{33} b^{24} \).

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Key Concepts

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

Power of a Product Rule
The power of a product rule is a nifty tool in algebra that makes exponentiation easy to handle when dealing with products. Whenever you see an expression like \((xy)^n\), this rule allows you to distribute the power to each factor: \((xy)^n = x^n y^n\).

This means you apply the exponent to every element inside the parentheses separately. So, if you take an expression such as \((a^{11} b^8)^3\), the rule helps you break it down:
  • Apply the exponent 3 to both \(a^{11}\) and \(b^8\).
  • This becomes \((a^{11})^3 \times (b^8)^3\).
By using the power of a product rule, you organize and simplify complex expressions right from the start, making it much easier to tackle the next steps.
Power of a Power Rule
Next in our toolkit is the power of a power rule. It's like magic for simplifying expressions where one power is raised to another power. This rule tells us that if you have \((x^m)^n\), you can rewrite it as \(x^{m \times n}\).

This simplification is crucial when you face something like \((a^{11})^3\) or \((b^8)^3\). You apply the rule as follows:
  • For \((a^{11})^3\), multiply the exponents: \(11 \times 3\), resulting in \(a^{33}\).
  • For \((b^8)^3\), multiply the exponents: \(8 \times 3\), resulting in \(b^{24}\).
By mastering the power of a power rule, you can handle nested exponents efficiently, turning them into clearly simplified forms.
Simplifying Expressions
Simplifying expressions is all about making complex expressions easier to work with or interpret. Once you apply the power of a product and power of a power rules, you often find yourself with an expression that can still be condensed.

For our example, after applying these two rules, we are left with \(a^{33} \times b^{24}\). This is achieved by:
  • Breaking down \((a^{11} b^8)^3\) into \((a^{11})^3 \times (b^8)^3\).
  • Simplifying each part: \((a^{11})^3 = a^{33}\) and \((b^8)^3 = b^{24}\).
The expression \(a^{33} b^{24}\) is the final, simplified result. It's easier to handle and see that each component's power is precise, making calculations or further algebraic manipulations straightforward.

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Most popular questions from this chapter

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