The Fully Simplified Form Of 2ab(8a) Is:A) [tex]\frac{2ab}{8a}[/tex] B) [tex]\frac{4}{b}[/tex] C) [tex]\frac{b}{4}[/tex]
Understanding how to simplify algebraic expressions is a fundamental skill in mathematics. In particular, simplifying expressions involving variables and coefficients helps to make complex problems more manageable and sets the foundation for higher-level math topics such as algebra, calculus, and beyond. Today, we will explore the expression 2ab(8a) and determine its fully simplified form among the provided options:
- A) \(\frac{2ab}{8a}\)
- B) \(\frac{4}{b}\)
- C) \(\frac{b}{4}\)
By the end of this article, you'll understand how to approach such problems systematically, use algebraic properties effectively, and confidently select the correct simplified expression.
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Understanding the Expression: 2ab(8a)
Before diving into simplification, it's essential to understand the components of the expression:
- 2ab: A coefficient (2) multiplied by variables \(a\) and \(b\).
- (8a): A coefficient (8) multiplied by variable \(a\).
When these are multiplied together, the expression becomes:
\[
2ab \times 8a
\]
The key to simplifying such an expression is to apply algebraic multiplication rules and combine like terms where possible.
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Step-by-Step Simplification Process
Let's break down the process systematically:
1. Write the Expression Explicitly
\[
2ab \times 8a
\]
This is a multiplication of two algebraic terms.
2. Apply the Associative and Commutative Properties
Multiplication of algebraic terms allows us to rearrange and regroup factors:
\[
(2 \times 8) \times (a \times a \times b)
\]
- Multiply the coefficients: \(2 \times 8 = 16\)
- Multiply variables: \(a \times a = a^2\)
Thus, the expression simplifies to:
\[
16 \times a^2 \times b
\]
or written as:
\[
16a^2b
\]
3. Recognize the Simplified Form
The simplified form of 2ab(8a) is 16a²b.
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Analyzing the Given Options
Now, let's analyze the options provided:
- Option A) \(\frac{2ab}{8a}\)
- Option B) \(\frac{4}{b}\)
- Option C) \(\frac{b}{4}\)
Our goal is to determine which of these options is equivalent to the fully simplified form 16a²b.
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Evaluating Each Option
Option A: \(\frac{2ab}{8a}\)
Let's simplify this fraction:
\[
\frac{2ab}{8a}
\]
- Divide numerator and denominator by common factors:
\[
\frac{2ab}{8a} = \frac{(2 \times a \times b)}{(8 \times a)}
\]
- Cancel common factors:
\[
\frac{2 \times \cancel{a} \times b}{8 \times \cancel{a}} = \frac{2b}{8}
\]
- Simplify numerator and denominator:
\[
\frac{2b}{8} = \frac{b}{4}
\]
So, Option A simplifies to \(\frac{b}{4}\), which is Option C.
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Option B: \(\frac{4}{b}\)
This is a simple fraction with numerator 4 and denominator \(b\). It does not resemble the fully simplified form \(16a^2b\), which involves \(a^2\) and a coefficient of 16. Since these do not match, Option B is not equivalent to the fully simplified expression.
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Option C: \(\frac{b}{4}\)
This was obtained after simplifying Option A, but it does not match the fully simplified form 16a^2b. It is a different expression altogether.
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Conclusion: Which Option Is Correct?
From the above analysis:
- Our fully simplified form of the original expression 2ab(8a) is 16a^2b.
- None of the options directly corresponds to 16a^2b.
- Option A, when simplified, results in \(\frac{b}{4}\), which does not match the fully simplified form.
- Option B (\(\frac{4}{b}\)) is unrelated.
- Option C (\(\frac{b}{4}\)) is also unrelated to the fully simplified form.
However, considering the options given and the process of simplification, the initial expression 2ab(8a) simplifies to 16a^2b, which does not match any of the options directly.
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Additional Insights and Clarifications
Why are the options provided potentially confusing?
Often in algebraic problems, options are designed to test understanding of simplification and algebraic manipulation. The options provided can represent different stages or types of expressions, but only one matches the fully simplified form.
In this case, the options seem to be designed to test the student's ability to manipulate fractions and variables.
Key Takeaways:
- When multiplying algebraic expressions, multiply coefficients and variables separately.
- Use the properties of exponents to simplify powers of variables.
- Cancel common factors when simplifying fractions.
- Always compare the fully simplified form to the options to determine the correct choice.
Summary and Final Remarks
- The expression 2ab(8a) simplifies to 16a^2b.
- Options A, B, and C are different algebraic expressions, with Option A simplifying to \(\frac{b}{4}\).
- None of the options directly matches the fully simplified form 16a^2b, indicating a possible mismatch or a trick question.
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Additional Resources for Learning Algebra Simplification
- Algebra Basics: Understanding variables, coefficients, and exponents.
- Simplifying Expressions: Techniques for expanding and reducing algebraic expressions.
- Fraction Simplification: Learning how to cancel common factors.
- Practice Problems: Applying these concepts to reinforce understanding.
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Remember: Practice makes perfect. The more you work with algebraic expressions, the more intuitive simplification becomes. Keep exploring different types of problems to strengthen your mathematical foundation!
