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Chapter 10: Problem 109

In the following exercises, simplify each expression. $$(5 a)^{2}(2 a)^{3}$$

Short Answer

Expert verified
The simplified expression is \( 200a^5 \).

Step by step solution

01

Apply the Power Rule

First, apply the power rule \(a^m \times a^n = a^{m+n}\). Simplify each term inside the parentheses separately: \( (5a)^2 = 5^2 \times a^2 = 25a^2 \) and \( (2a)^3 = 2^3 \times a^3 = 8a^3 \).
02

Multiply the Coefficients

Next, multiply the numerical coefficients together: \( 25 \times 8 = 200 \).
03

Use the Power Rule for the Variables

Combine the powers of \(a\) using \(a^m \times a^n = a^{m+n}\). Here, \( a^2 \times a^3 = a^{2+3} = a^5 \).
04

Combine the Results

Combine the results of the previous steps to get the simplified expression: \( 200a^5 \).

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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 in algebra helps us manage expressions with exponents. It states that when you multiply two exponents with the same base, you add the exponents: \(a^m \times a^n = a^{m+n}\).
This rule is crucial when we simplify expressions involving variables raised to powers. It ensures our calculations are concise and correct.
Let's look at an example: if we have \((5a)^2 (2a)^3\), first we deal with the parentheses. We apply the power to each element inside:
\((5a)^2 = 5^2 \times a^2\) and \((2a)^3 = 2^3 \times a^3\).
These steps are foundational and set us up for accurately simplifying the expression.
Multiplying Coefficients
Once we've applied the power rule, we focus on the numerical parts, or coefficients, of our expression. Multiplying coefficients is straightforward:
  • First, simplify any constants raised to a power. For instance, \(5^2 = 25\) and \(2^3 = 8\).
  • Next, multiply these values together: \(25 \times 8 = 200\).
This step results in the simplified coefficient of the final expression.
The coefficients should always be simplified before tackling the variable parts, ensuring we do not mix numerical and variable rules.
Combining Exponents
After multiplying the coefficients, we look at the variables. Similar to coefficients, we apply the power rule again, but only to the variable part:
  • If we have \(a^2 \times a^3\), we add the exponents: \(a^{2+3} = a^5\).
This rule helps simplify and combine variables raised to different powers, making large expressions manageable. Remember, the base must be the same for this rule to apply.
Prealgebra
Prealgebra covers the foundational concepts you need for more advanced math.
It includes understanding variables, basic rules of exponents, and simplifying expressions. Developing these skills is essential for confidently tackling algebra.
  • Knowing how to simplify algebraic expressions brings clarity and ease.
  • These basic operations, like applying the power rule and multiplying coefficients, are the building blocks of algebra.
  • The more you practice these rules, the more intuitive they become, paving the way for mastering complex problems.
Regular practice and step-by-step problem-solving ensure you grasp these fundamental concepts thoroughly.

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