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Chapter 10: Problem 70

Simplify. $$\left(p^{44} z^{32}\right)^{3}$$

Short Answer

Expert verified
p^{132} z^{96}

Step by step solution

01

Identify the Expression

The given expression is \((p^{44} z^{32})^{3}\). This is a power of a product.
02

Use the Power of a Power Rule

The power of a power rule states that \((a^m)^n = a^{mn}\). Apply this rule to each term in the expression.
03

Simplify the Exponent of p

Apply the rule to \(p^{44}\): \((p^{44})^3 = p^{44 \times 3} = p^{132}\).
04

Simplify the Exponent of z

Apply the rule to \(z^{32}\): \((z^{32})^3 = z^{32 \times 3} = z^{96}\).
05

Combine the Results

Combine the simplified terms: \(p^{132} z^{96}\).

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Key Concepts

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

Power of a Power Rule
When you see an expression like \( (a^m)^n \), you can use the power of a power rule to simplify it. This rule tells you to multiply the exponents together. For example, \( (p^{44})^3 \) can be simplified by multiplying the exponents: \( 44 \times 3 = 132\). This gives us \( p^{132} \). The same goes for \( (z^{32})^3 \), which simplifies to \( z^{96} \).

This rule works because of how exponents represent repeated multiplication. If you raise a power to another power, you're essentially combining all of those repeated multiplications.
Simplifying Expressions
Simplifying expressions involves breaking down complicated expressions into simpler, more manageable parts. By following rules like the power of a power rule, you can easily simplify complex expressions. For the given problem \( (p^{44} z^{32})^3 \), we applied the power of a power rule to each term individually before combining them.

Here are the steps for simplifying:
  • Identify the expression and break it down into its components.
  • Apply appropriate rules (e.g., power of a power).
  • Combine the results to get the simplified expression.
Following these steps will help you to simplify any similar expressions efficiently.
Exponents Multiplication
Exponents multiplication involves understanding how to handle expressions that require multiplying powers. In the given problem, we multiplied exponents as follows: \( (p^{44})^3 = p^{44 \times 3} \), which resulted in \( p^{132} \). Similarly, \( (z^{32})^3 = z^{32 \times 3} = z^{96} \).

To multiply exponents when raising a power to another power, follow these steps:
  • Identify the base and the exponents you need to multiply.
  • Multiply the exponents together.
  • Write the new expression with the multiplied exponent applied to the base.
This is a crucial skill for simplifying and manipulating expressions involving exponents.

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Most popular questions from this chapter

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