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Chapter 10: Problem 41

Simplify the expression. $$(2 w)^{5}$$

Short Answer

Expert verified
The simplified expression is \(32w^5\).

Step by step solution

01

- Understand the Problem Statement

Given the expression \( (2w)^5 \), the goal is to simplify it. This involves applying the properties of exponents correctly.
02

- Apply the Power of a Product Rule

The Power of a Product Rule states that \( (ab)^n = a^n \cdot b^n \). Apply this rule to \( (2w)^5 \): \( (2w)^5 = 2^5 \cdot w^5 \).
03

- Calculate the Exponents

Calculate \(2^5\) and combine it with \(w^5\): \( 2^5 = 32 \). Therefore, \( 2^5 \cdot w^5 = 32w^5 \).

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Key Concepts

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

properties of exponents
When working with expressions that contain exponents, it鈥檚 important to understand the underlying properties. Exponents represent repeated multiplication. For example, in the term \(a^n\), 'a' is the base and 'n' is the exponent. This indicates that 'a' is multiplied by itself 'n' times.
Here are some key properties:
  • Product of Powers Property: \(a^m \cdot a^n = a^{m+n}\).
  • Power of a Power Property: \((a^m)^n = a^{m \times n}\).
  • Power of a Product Property: \((ab)^n = a^n \cdot b^n\).
  • Quotient of Powers Property: \(\frac{a^m}{a^n} = a^{m-n}\), provided \(a eq 0\).
Understanding these properties is essential for simplifying expressions and solving equations effectively.
power of a product rule
One of the most useful properties of exponents is the Power of a Product Rule, which is particularly relevant for this exercise. This rule states that \((ab)^n = a^n \cdot b^n\). In simpler terms:
  • When you have a product inside parentheses raised to a power, you can distribute the power to each factor in the product individually.
  • This rule simplifies calculations and helps in breaking down complex expressions into manageable parts.
Let鈥檚 apply this rule to the given expression: \( (2w)^5 \). Applying the Power of a Product Rule, we get \( (2w)^5 = 2^5 \cdot w^5 \). Now, each part can be calculated separately.
calculating exponents
Once the Power of a Product Rule has been applied, the next step is to calculate the exponents individually. In our example, we need to calculate \(2^5\) and retain \(w^5\):
  • First, calculate \(2^5\). Since \(2^5\) is the same as multiplying 2 by itself five times: \(2 \times 2 \times 2 \times 2 \times 2 = 32\).
  • The variable part, \(w^5\), remains as it is because we don鈥檛 have a specific value for 'w'.
This gives us the simplified expression: \(32w^5\).
The steps for calculating exponents are straightforward once you understand the properties. Practice applying these rules to different problems to become more confident.

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