Which Expression Is Equivalent To $\frac{\left(12 A^3\right)(3 A)}{\left(9 A^2\right)}$?A. $\frac{15 A^2}{9}$ B. \$4 A$[/tex\] C. $4 A^2$ D. $6 A^2$
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Introduction
Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the process of simplifying algebraic expressions, with a focus on the given expression $\frac{\left(12 a^3\right)(3 a)}{\left(9 a^2\right)}$. We will break down the expression into smaller parts, apply the rules of exponents, and simplify the resulting expression.
Understanding the Given Expression
The given expression is $\frac{\left(12 a^3\right)(3 a)}{\left(9 a^2\right)}$. This expression involves multiplication and division of variables and constants. To simplify this expression, we need to apply the rules of exponents and follow the order of operations (PEMDAS).
Applying the Rules of Exponents
The first step in simplifying the expression is to apply the rules of exponents. When multiplying variables with the same base, we add their exponents. In this case, we have $a^3 \cdot a = a^{3+1} = a^4$.
Simplifying the Numerator
The numerator of the expression is $\left(12 a^3\right)(3 a)$. We can simplify this by multiplying the constants and adding the exponents of the variables. This gives us $36 a^4$.
Simplifying the Denominator
The denominator of the expression is $\left(9 a^2\right)$. We can simplify this by multiplying the constants and adding the exponents of the variables. This gives us $9 a^2$.
Simplifying the Expression
Now that we have simplified the numerator and denominator, we can simplify the expression by dividing the numerator by the denominator. This gives us $\frac{36 a^4}{9 a^2}$.
Canceling Common Factors
We can simplify the expression further by canceling common factors. In this case, we can cancel $9$ from the numerator and denominator, and $a^2$ from the numerator and denominator. This gives us $4 a^2$.
Conclusion
In conclusion, the simplified expression is $4 a^2$. This is the correct answer among the given options.
Final Answer
The final answer is: C. $4 a^2$
Discussion
The given expression involves multiplication and division of variables and constants. To simplify this expression, we need to apply the rules of exponents and follow the order of operations (PEMDAS). The key steps in simplifying the expression are:
- Applying the rules of exponents to simplify the numerator and denominator
- Canceling common factors to simplify the expression further
By following these steps, we can simplify the given expression and arrive at the correct answer.
Related Topics
- Algebraic expressions
- Simplifying expressions
- Rules of exponents
- Order of operations (PEMDAS)
Further Reading
For more information on simplifying algebraic expressions, check out the following resources:
- Khan Academy: Simplifying Expressions
- Mathway: Simplifying Algebraic Expressions
- Wolfram Alpha: Simplifying Algebraic Expressions
FAQs
- Q: What is the simplified expression for $\frac{\left(12 a^3\right)(3 a)}{\left(9 a^2\right)}$? A: The simplified expression is $4 a^2$.
- Q: How do I simplify an algebraic expression? A: To simplify an algebraic expression, apply the rules of exponents and follow the order of operations (PEMDAS).
- Q: What are the key steps in simplifying an algebraic expression?
A: The key steps are applying the rules of exponents, canceling common factors, and following the order of operations (PEMDAS).
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Introduction
Simplifying algebraic expressions is a crucial skill in mathematics, and it can be a bit challenging for beginners. In this article, we will address some of the most frequently asked questions about simplifying algebraic expressions.
Q&A
Q: What is an algebraic expression?
A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations. It is a way to represent a mathematical relationship between variables and constants.
Q: How do I simplify an algebraic expression?
A: To simplify an algebraic expression, you need to apply the rules of exponents and follow the order of operations (PEMDAS). This involves simplifying the numerator and denominator separately and then simplifying the resulting expression.
Q: What are the rules of exponents?
A: The rules of exponents are a set of rules that govern the behavior of exponents in algebraic expressions. The main rules are:
- When multiplying variables with the same base, add their exponents.
- When dividing variables with the same base, subtract their exponents.
- When raising a variable to a power and then raising it to another power, multiply the exponents.
Q: What is the order of operations (PEMDAS)?
A: The order of operations (PEMDAS) is a set of rules that govern the order in which mathematical operations are performed in an algebraic expression. The acronym PEMDAS stands for:
- Parentheses: Evaluate expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
Q: How do I simplify a fraction with variables?
A: To simplify a fraction with variables, you need to simplify the numerator and denominator separately and then simplify the resulting fraction. This involves applying the rules of exponents and following the order of operations (PEMDAS).
Q: What is the difference between a variable and a constant?
A: A variable is a symbol that represents a value that can change, while a constant is a value that remains the same.
Q: How do I simplify an expression with multiple variables?
A: To simplify an expression with multiple variables, you need to apply the rules of exponents and follow the order of operations (PEMDAS). This involves simplifying the numerator and denominator separately and then simplifying the resulting expression.
Q: What are some common mistakes to avoid when simplifying algebraic expressions?
A: Some common mistakes to avoid when simplifying algebraic expressions include:
- Not following the order of operations (PEMDAS)
- Not applying the rules of exponents correctly
- Not simplifying the numerator and denominator separately
- Not canceling common factors
Conclusion
Simplifying algebraic expressions is a crucial skill in mathematics, and it can be a bit challenging for beginners. By following the rules of exponents and the order of operations (PEMDAS), you can simplify even the most complex algebraic expressions.
Final Tips
- Practice simplifying algebraic expressions regularly to build your skills and confidence.
- Use online resources and tools to help you simplify algebraic expressions.
- Don't be afraid to ask for help if you're struggling with a particular expression.
Related Topics
- Algebraic expressions
- Simplifying expressions
- Rules of exponents
- Order of operations (PEMDAS)
Further Reading
For more information on simplifying algebraic expressions, check out the following resources:
- Khan Academy: Simplifying Expressions
- Mathway: Simplifying Algebraic Expressions
- Wolfram Alpha: Simplifying Algebraic Expressions
FAQs
- Q: What is the simplified expression for $\frac{\left(12 a^3\right)(3 a)}{\left(9 a^2\right)}$? A: The simplified expression is $4 a^2$.
- Q: How do I simplify an algebraic expression? A: To simplify an algebraic expression, apply the rules of exponents and follow the order of operations (PEMDAS).
- Q: What are the key steps in simplifying an algebraic expression? A: The key steps are applying the rules of exponents, canceling common factors, and following the order of operations (PEMDAS).