Problem 110 In the following exercises, simp... [FREE SOLUTION]

91影视

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 110

In the following exercises, simplify each expression. $$(4 b)^{2}(3 b)^{3}$$

Short Answer

Expert verified

432b^5

Step by step solution

Identify the expression's components

The expression to simplify is $ (4b)^2 (3b)^3 $.

Apply the power rule

Use the power rule, which states $ (a^m b^n)^p = a^{mp} b^{np} $, to simplify each term separately: $ (4b)^2 = 4^2 \times b^2 = 16b^2 $ and $ (3b)^3 = 3^3 \times b^3 = 27b^3 $.

Multiply the simplified terms

Now multiply the simplified terms: $ 16b^2 \times 27b^3 $.

Combine like terms

Multiply the coefficients and add the exponents of the bases: $ 16 \times 27 = 432 $ and $ b^2 \times b^3 = b^{2+3} = b^5 $.

Simplify the final expression

Combine all parts to get the final simplified expression: $ 432b^5 $.

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Key Concepts

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

Power Rule

The power rule is essential in algebra for simplifying expressions where variables are raised to a power. This rule states that for any base 'a' with exponents 'm' and 'p', $ (a^m)^p = a^{m \cdot p} $. For example, if we have $ (4b)^2 $, it means both the number 4 and the variable 'b' are raised to the power of 2. This gives us:
\[ (4b)^2 = 4^2 \cdot b^2 = 16b^2 \].
Similarly, $(3b)^3$ is computed as: \[ (3b)^3 = 3^3 \cdot b^3 = 27b^3 \].
By splitting the coefficient and the variable, and then raising each to the given power, we make the expression simpler to manage.

Multiplying Expressions

When multiplying expressions in algebra, we combine both the numerical coefficients and the variable parts separately. Let's multiply $16b^2 $ and $ 27b^3 $. We start with the coefficients: 16 and 27.
Multiply them: \[ 16 \times 27 = 432 \].
Then, handle the variable parts. Since both terms have 'b' as the base, we add their exponents: \[ b^2 \cdot b^3 = b^{2+3} = b^5 \].
So the product of $ 16b^2 $ and $ 27b^3 $ is $ 432b^5 $.
Breaking it down:

Multiply coefficients together.
Add exponents of like bases.

This makes combining and simplifying expressions straightforward.

Combining Like Terms

Combining like terms means adding or subtracting terms with the same variable raised to the same power. In our expression, we simplified and multiplied terms with the same base 'b'. The same idea applies when simplifying any algebraic expression.
For example, in the step: \[ (4b)^2 (3b)^3 = 16b^2 \cdot 27b^3 \],
we multiplied the coefficients to get 432, and combined the \b \ terms by adding their exponents, resulting in \ b^{2+3} = b^5 \.
Always remember:

Combine coefficients by arithmetic operations.
Add exponents for like variables.

This results in a simplified and more manageable expression, like our final answer: \ 432b^5 \.

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91影视

In the following exercises, simplify each expression. $$(4 b)^{2}(3 b)^{3}$$

Short Answer

Step by step solution

Identify the expression's components

Apply the power rule

Multiply the simplified terms

Combine like terms

Simplify the final expression

Key Concepts

Power Rule

Multiplying Expressions

Combining Like Terms

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