Evaluate The Expression:${ 6 \times \frac{\sqrt{3}}{2} }$
Introduction
In mathematics, evaluating expressions is a crucial skill that involves simplifying mathematical statements to obtain a numerical value. In this article, we will focus on evaluating the expression 6 × √3/2, which involves multiplication and division of fractions. We will break down the expression step by step, using mathematical concepts and formulas to simplify it.
Understanding the Expression
The given expression is 6 × √3/2. To evaluate this expression, we need to understand the order of operations, which is a set of rules that dictate the order in which mathematical operations should be performed. The order of operations is often remembered using the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
In this expression, we have multiplication and division, but no parentheses or exponents. Therefore, we will follow the order of operations and perform the multiplication and division from left to right.
Multiplying 6 by √3
To evaluate the expression 6 × √3/2, we need to start by multiplying 6 by √3. The square root of 3 (√3) is an irrational number that is approximately equal to 1.732. Therefore, we can write the expression as:
6 × 1.732
To multiply 6 by 1.732, we can simply multiply the two numbers:
6 × 1.732 = 10.392
Dividing by 2
Now that we have multiplied 6 by √3, we need to divide the result by 2. To do this, we can simply divide 10.392 by 2:
10.392 ÷ 2 = 5.196
Conclusion
In conclusion, the expression 6 × √3/2 can be evaluated by multiplying 6 by √3 and then dividing the result by 2. Using the order of operations and mathematical concepts, we can simplify the expression to obtain a numerical value of 5.196.
Simplifying the Expression Using Mathematical Concepts
In the previous section, we evaluated the expression 6 × √3/2 by multiplying 6 by √3 and then dividing the result by 2. However, we can also simplify the expression using mathematical concepts.
One way to simplify the expression is to use the fact that √3 is an irrational number that can be expressed as a decimal. We can write √3 as 1.732, and then multiply 6 by this decimal:
6 × 1.732 = 10.392
However, we can also simplify the expression by using the fact that √3 is a root of the equation x^2 - 3 = 0. This means that √3 is a number that, when squared, gives 3. We can use this fact to simplify the expression:
6 × √3/2 = 6 × (√3)^2 / 2
Using the fact that (√3)^2 = 3, we can simplify the expression:
6 × (√3)^2 / 2 = 6 × 3 / 2
Now, we can simplify the expression by multiplying 6 by 3 and then dividing the result by 2:
6 × 3 = 18
18 ÷ 2 = 9
Using Trigonometry to Simplify the Expression
In the previous section, we simplified the expression 6 × √3/2 by using mathematical concepts. However, we can also simplify the expression using trigonometry.
One way to simplify the expression is to use the fact that √3 is related to the sine and cosine functions. Specifically, we can use the fact that sin(60°) = √3/2. This means that √3 is equal to 2 times the sine of 60°.
We can use this fact to simplify the expression:
6 × √3/2 = 6 × (2 × sin(60°)) / 2
Using the fact that sin(60°) = √3/2, we can simplify the expression:
6 × (2 × sin(60°)) / 2 = 6 × 2 × (√3/2) / 2
Now, we can simplify the expression by canceling out the 2's:
6 × 2 × (√3/2) / 2 = 6 × (√3/2)
Using the fact that sin(60°) = √3/2, we can simplify the expression:
6 × (√3/2) = 6 × sin(60°)
Now, we can simplify the expression by using the fact that sin(60°) = √3/2:
6 × sin(60°) = 6 × (√3/2)
Using the fact that sin(60°) = √3/2, we can simplify the expression:
6 × (√3/2) = 6 × (√3/2)
Now, we can simplify the expression by canceling out the 2's:
6 × (√3/2) = 3 × √3
Using the fact that √3 is an irrational number that can be expressed as a decimal, we can simplify the expression:
3 × √3 = 3 × 1.732
Now, we can simplify the expression by multiplying 3 by 1.732:
3 × 1.732 = 5.196
Conclusion
In conclusion, the expression 6 × √3/2 can be evaluated by multiplying 6 by √3 and then dividing the result by 2. Using mathematical concepts and trigonometry, we can simplify the expression to obtain a numerical value of 5.196.
Final Answer
The final answer to the expression 6 × √3/2 is 5.196.
Introduction
In our previous article, we evaluated the expression 6 × √3/2 using mathematical concepts and trigonometry. In this article, we will answer some frequently asked questions (FAQs) related to the expression.
Q: What is the value of √3?
A: The value of √3 is an irrational number that is approximately equal to 1.732.
Q: How do I simplify the expression 6 × √3/2?
A: To simplify the expression 6 × √3/2, you can multiply 6 by √3 and then divide the result by 2. Alternatively, you can use mathematical concepts and trigonometry to simplify the expression.
Q: What is the order of operations for evaluating the expression 6 × √3/2?
A: The order of operations for evaluating the expression 6 × √3/2 is as follows:
- Multiply 6 by √3
- Divide the result by 2
Q: Can I use trigonometry to simplify the expression 6 × √3/2?
A: Yes, you can use trigonometry to simplify the expression 6 × √3/2. Specifically, you can use the fact that sin(60°) = √3/2 to simplify the expression.
Q: What is the final answer to the expression 6 × √3/2?
A: The final answer to the expression 6 × √3/2 is 5.196.
Q: Can I use a calculator to evaluate the expression 6 × √3/2?
A: Yes, you can use a calculator to evaluate the expression 6 × √3/2. However, it's also important to understand the mathematical concepts and trigonometry behind the expression.
Q: How do I apply the order of operations to evaluate the expression 6 × √3/2?
A: To apply the order of operations to evaluate the expression 6 × √3/2, follow these steps:
- Multiply 6 by √3
- Divide the result by 2
For example, if you multiply 6 by √3, you get:
6 × √3 = 6 × 1.732 = 10.392
Then, you divide the result by 2:
10.392 ÷ 2 = 5.196
Q: Can I use the fact that √3 is a root of the equation x^2 - 3 = 0 to simplify the expression 6 × √3/2?
A: Yes, you can use the fact that √3 is a root of the equation x^2 - 3 = 0 to simplify the expression 6 × √3/2. Specifically, you can use the fact that (√3)^2 = 3 to simplify the expression.
Q: How do I use trigonometry to simplify the expression 6 × √3/2?
A: To use trigonometry to simplify the expression 6 × √3/2, follow these steps:
- Use the fact that sin(60°) = √3/2 to simplify the expression
- Multiply 6 by sin(60°)
- Divide the result by 2
For example, if you multiply 6 by sin(60°), you get:
6 × sin(60°) = 6 × (√3/2) = 3 × √3
Then, you divide the result by 2:
3 × √3 ÷ 2 = 5.196
Conclusion
In this article, we answered some frequently asked questions (FAQs) related to the expression 6 × √3/2. We hope that this article has provided you with a better understanding of the expression and how to evaluate it using mathematical concepts and trigonometry.