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Chapter 9: Problem 80

Simplify. $$\left(-3 b^{2}\right)\left(2 a b^{2}\right)^{3}$$

Short Answer

Expert verified
The simplified expression is \(-24a^{3}b^{8}\).

Step by step solution

01

Distribution of power

Distribute the power of 3 inside the brackets to each term. That is, \( (2ab^{2})^{3} = 2^{3} * a^{3} * b^{2*3} = 8a^{3}b^{6} \).
02

Multiplication

Multiply \(-3b^{2}\) by \(8a^{3}b^{6}\) by multiplying the numerical values and adding the powers of like bases. So, \(-3b^{2} * 8a^{3}b^{6} = -24a^{3}b^{8}\).
03

Simplification

The expression \(-24a^{3}b^{8}\) is the simplified form, which cannot be minimized further.

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Key Concepts

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

Distributive Property
The distributive property is a core mathematical principle used frequently in algebra to simplify expressions. It allows you to multiply a term across terms within parentheses. For example, if you have an expression like \((a + b)c\), you can distribute the \(c\) to both \(a\) and \(b\), leading to \(ac + bc\).

In the exercise provided, the distributive property is not directly applied in its basic sense, as it focuses more on distributing an exponent over a product. However, understanding distribution helps grasp why each term, such as \(2a\) and \(b^{2}\), in the parenthesis \((2ab^{2})\) must be raised to the power separately. Remember:
  • Distribution can apply to multiplication over addition or in applying powers to products.
  • It simplifies complex expressions into fundamental calculations.
Exponent Rules
Exponent rules are essential for simplifying expressions involving powers. They help dictate how to handle exponents during multiplication, division, or when raising powers to powers. These rules include the power of a power rule, which states that when raising a power to another power, you multiply the exponents.

In our example, \((2ab^{2})^{3}\) uses this very principle. First, distribute the exponent to each term inside the parentheses:
  • The coefficient \(2^3\) results in \(8\).
  • The variable \(a\) becomes \(a^{3}\).
  • The exponent 2 on \(b^{2}\) is multiplied by 3: \(b^{2*3} = b^{6}\).
After distributing the exponent, the expression becomes \(8a^3b^6\). This is a crucial step in simplifying and requires a strong grasp of exponent rules.

These rules save time, minimize errors, and ensure accurate simplification.
Algebraic Expressions
Algebraic expressions include numbers, variables, and operations combined to represent a value or equation. These expressions can be simplified by using properties and rules, such as the distributive property and exponent rules, to reduce them to simpler forms.

In the given exercise, we start with a complex expression: \((-3b^{2})(2ab^{2})^{3}\). Breaking it down step-by-step, we manipulate and simplify the expression while respecting mathematical laws and properties.
  • Expression \((2ab^{2})^{3}\) simplifies using exponent rules.
  • Then it is multiplied by \(-3b^{2}\) by multiplying constants and adding exponents of like terms.
This yields \(-24a^{3}b^{8}\), a fully simplified algebraic expression. Understanding how to handle such expressions is crucial in algebra for solving equations and rearranging formulas.

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