Which Is Equal To $4 \sqrt{9} + 14 \sqrt{4}$?A. 46 B. 50 C. 40 D. 64
Introduction
Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying radical expressions, with a focus on the given expression $4 \sqrt{9} + 14 \sqrt{4}$. We will break down the expression into its individual components, simplify each radical, and then combine the results to find the final answer.
Understanding Radical Expressions
A radical expression is a mathematical expression that contains a square root or a higher-order root. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16.
Simplifying the Given Expression
The given expression is $4 \sqrt{9} + 14 \sqrt{4}$. To simplify this expression, we need to start by simplifying each radical individually.
Simplifying $\sqrt{9}$
The square root of 9 is 3, because 3 multiplied by 3 equals 9. Therefore, $\sqrt{9} = 3$.
Simplifying $\sqrt{4}$
The square root of 4 is 2, because 2 multiplied by 2 equals 4. Therefore, $\sqrt{4} = 2$.
Substituting the Simplified Radicals
Now that we have simplified each radical, we can substitute the results back into the original expression.
Evaluating the Expression
To evaluate the expression, we need to follow the order of operations (PEMDAS):
- Multiply 4 and 3: 4(3) = 12
- Multiply 14 and 2: 14(2) = 28
- Add 12 and 28: 12 + 28 = 40
Therefore, the simplified expression is $40$.
Conclusion
In this article, we simplified the radical expression $4 \sqrt{9} + 14 \sqrt{4}$ by breaking it down into its individual components, simplifying each radical, and then combining the results. We found that the simplified expression is equal to $40$. This process demonstrates the importance of simplifying radical expressions in mathematics and provides a step-by-step guide for solving similar problems.
Final Answer
The final answer is: 40
Comparison of Options
Let's compare our final answer with the given options:
- A. 46: This is not the correct answer.
- B. 50: This is not the correct answer.
- C. 40: This is the correct answer.
- D. 64: This is not the correct answer.
Introduction
In our previous article, we explored the process of simplifying radical expressions, with a focus on the given expression $4 \sqrt{9} + 14 \sqrt{4}$. We broke down the expression into its individual components, simplified each radical, and then combined the results to find the final answer. In this article, we will answer some frequently asked questions related to simplifying radical expressions.
Q&A
Q: What is the difference between a radical and a rational number?
A: A radical is a mathematical expression that contains a square root or a higher-order root, while a rational number is a number that can be expressed as the ratio of two integers.
Q: How do I simplify a radical expression?
A: To simplify a radical expression, you need to start by simplifying each radical individually. This involves finding the square root of the number inside the radical sign and then simplifying the resulting expression.
Q: What is the order of operations for simplifying radical expressions?
A: The order of operations for simplifying radical expressions is:
- Simplify each radical individually
- Combine like terms
- Evaluate any exponential expressions
- Perform any necessary arithmetic operations
Q: Can I simplify a radical expression with a variable inside the radical sign?
A: Yes, you can simplify a radical expression with a variable inside the radical sign. However, you need to follow the same steps as before: simplify each radical individually, combine like terms, and then evaluate any exponential expressions.
Q: How do I know if a radical expression can be simplified?
A: A radical expression can be simplified if the number inside the radical sign is a perfect square or a perfect cube. If the number is not a perfect square or perfect cube, the radical expression cannot be simplified.
Q: Can I simplify a radical expression with a negative number inside the radical sign?
A: Yes, you can simplify a radical expression with a negative number inside the radical sign. However, you need to follow the same steps as before: simplify each radical individually, combine like terms, and then evaluate any exponential expressions.
Q: What is the difference between a radical and an exponent?
A: A radical is a mathematical expression that contains a square root or a higher-order root, while an exponent is a mathematical expression that represents repeated multiplication of a number.
Q: Can I simplify a radical expression with a fraction inside the radical sign?
A: Yes, you can simplify a radical expression with a fraction inside the radical sign. However, you need to follow the same steps as before: simplify each radical individually, combine like terms, and then evaluate any exponential expressions.
Q: How do I know if a radical expression is equal to a rational number?
A: A radical expression is equal to a rational number if the number inside the radical sign is a perfect square or a perfect cube. If the number is not a perfect square or perfect cube, the radical expression is not equal to a rational number.
Conclusion
In this article, we answered some frequently asked questions related to simplifying radical expressions. We covered topics such as the difference between a radical and a rational number, the order of operations for simplifying radical expressions, and how to simplify radical expressions with variables, negative numbers, and fractions inside the radical sign. We hope that this article has provided you with a better understanding of simplifying radical expressions and has helped you to answer some of the most common questions related to this topic.
Final Answer
The final answer is: 40
Comparison of Options
Let's compare our final answer with the given options:
- A. 46: This is not the correct answer.
- B. 50: This is not the correct answer.
- C. 40: This is the correct answer.
- D. 64: This is not the correct answer.
Therefore, the correct answer is C. 40.