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Chapter 5: Problem 16

Simplify each expression using the products-to-powers rule. $$(4 x)^{3}$$

Short Answer

Expert verified
The simplified expression is \(64x^3\).

Step by step solution

01

Identify the base and exponent

Firstly, identify the base and the exponent in the expression. The base is \(4x\) and the exponent is 3.
02

Apply the products-to-powers rule

Using the products-to-powers rule, this expression can be rewritten as \(4^3 * x^3\).
03

Calculate the value of the powers

Next, perform the calculations for each part. So, \(4^3\) equals 64 and \(x^3\) stays as it is. Thus, the expression simplifies to \(64x^3\).

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Key Concepts

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

Exponentiation
Exponentiation is a fundamental concept in algebra and mathematics as a whole. It involves raising a base number or expression to a power, or exponent. This tells you how many times to multiply the base by itself. For example, in the expression \( (4x)^3 \), the base is \( 4x \) and the exponent is 3, meaning the base is used as a factor three times: \( (4x) \times (4x) \times (4x) \).
Exponentiation helps in simplifying expressions and solving equations efficiently, especially when dealing with large numbers. It is crucial to understand the role the exponent plays in determining the magnitude of an expression.
Products-to-Powers Rule
The Products-to-Powers Rule is a handy tool when simplifying expressions that include powers. It's a specific rule of exponents that tells us how to handle expressions where a product inside parentheses is raised to an exponent.
  • When an expression like \((a \, b)^n\) appears, the rule states it can be expanded to \(a^n \, b^n\). This means each part of the product is raised to the power individually.
  • For the exercise \((4x)^3\), we apply this rule to expand it to \(4^3 \times x^3\).
Using this rule simplifies complex expressions into manageable parts, breaking them down into individually solvable units. Understanding and applying this rule is essential for efficiently tackling algebraic expressions involving products and exponents.
Simplification
Simplification of algebraic expressions involves reducing them to their simplest form. This often means performing calculations and rewriting expressions to be more concise and easier to interpret. In our exercise, after applying the Products-to-Powers Rule, the expression \(4^3 \times x^3\) is further simplified.
  • Calculate the powers: \(4^3\), which means multiplying 4 by itself three times, results in 64.
  • The expression \(x^3\) stays as it is since it doesn't numerically simplify further.
This simplification results in \(64x^3\). Understanding how to simplify expressions is key to effective algebraic problem-solving, allowing you to present results cleanly and clearly.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{x^{2}+x}{x}=x$$

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