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Chapter 5: Problem 19

Simplify each expression using the products to-powers rule. $$\left(4 x^{3}\right)^{2}$$

Short Answer

Expert verified
The simplified expression for \((4x^3)^2\) is \(16x^6\).

Step by step solution

01

Apply the product-to-powers rule

The product-to-powers rule states that \((ab)^n = a^n * b^n\). Apply this rule to \((4x^3)^2\). This means that both the integer 4 and the term \(x^3\) are raised to the power of 2.
02

Simplify the terms individually

Calculate \(4^2\) and \((x^3)^2\) individually. \(4^2 = 16\) and \((x^3)^2 = x^{3*2} = x^6\).
03

Combine the simplified terms

Combine the results of \(4^2\) and \((x^3)^2\) to get the final simplified expression. Thus, \((4x^3)^2 = 16x^6\).

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Key Concepts

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

Product-to-powers rule
The product-to-powers rule is a useful tool in simplifying algebraic expressions. It essentially breaks down the process of raising a product to a power. When you have an expression like \((ab)^n\), the rule allows you to distribute the exponent \(n\) to both \(a\) and \(b\). This simplifies the expression into two separate parts as \(a^n \times b^n\).

In the exercise \((4x^3)^2\), the product-to-powers rule helps us handle both the coefficient \(4\) and the variable part \(x^3\) separately.
  • First, apply the exponent 2 to the coefficient (4), resulting in \(4^2\).
  • Then, apply the exponent 2 to the variable expression \(x^3\), giving \((x^3)^2\).
By using the product-to-powers rule, each component is managed individually, making the expression easier to handle and ultimately simplify.
Exponentiation
Exponentiation is a fundamental concept in algebra where a number or an expression is raised to a power. This implies repeated multiplication of the base itself, according to the value of the exponent. For instance, \(4^2\) means 4 is multiplied by itself, resulting in 16. Similarly, \((x^3)^2\) involves raising \(x^3\) to the power of 2.

This can be seen as a multiplication of exponents, calculated as \(x^{3 \times 2} = x^6\). Here, exponentiation significantly reduces the complexity of expressions by summarizing repeated multiplications.
  • The base is 4 in the first example, and the exponent is 2.
  • In the case of \(x^3\), the base is \(x\) and the initial exponent is 3, which is further raised again by 2.
Understanding how exponentiation works simplifies complex calculations and is essential when dealing with polynomial expressions.
Polynomial expressions
Polynomial expressions are mathematical expressions involving a sum of terms, each consisting of a variable raised to an integer power, multiplied by a coefficient. These expressions can include one or multiple variables that may be raised to various powers. For example, a simple polynomial expression is \(4x^3\).

Polynomials often require simplification, which involves combining like terms and applying algebraic rules, such as the product-to-powers rule and exponentiation.
  • Simplifying \((4x^3)^2\) involves treating both the coefficient (4) and the power of variable (\(x^3\)) separately.
  • Each part is first raised to the required power before combining them back into a single, simplified term.
  • In this example, you would simplify \(4^2\) to get 16, and \(x^{3 \times 2}\), resulting in \(x^6\).
Simplifying polynomial expressions helps in organizing and solving complex algebraic problems in a more manageable format.

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Most popular questions from this chapter

Explain how to divide a polynomial that is not a monomial by a monomial. Give an example.

Explain the product rule for exponents. Use \(2^{3} \cdot 2^{5}\) in your explanation.

In Exercises \(79-82,\) simplify each expression. $$\frac{2 x^{3}(4 x+2)-3 x^{2}(2 x-4)}{2 x^{2}}$$

Explain how to multiply polynomials when neither is a monomial. Give an example.

In each exercise, find the product. $$(x+3)\left(x^{2}+5\right)$$
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