Evaluate The Expression: 3 8 6 2 + 8 2 + ( 1.5 ) 2 \frac{3}{8} \sqrt{6^2+8^2} + (1.5)^2 8 3 ​ 6 2 + 8 2 ​ + ( 1.5 ) 2

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Introduction

In mathematics, expressions involving square roots and fractions can be challenging to simplify. However, with a clear understanding of the concepts and a step-by-step approach, we can evaluate such expressions with ease. In this article, we will focus on simplifying the given expression: $\frac{3}{8} \sqrt{6^2+8^2} + (1.5)^2$. We will break down the solution into manageable steps, making it easy to follow and understand.

Understanding the Expression

The given expression involves a square root and a fraction. To simplify it, we need to start by evaluating the terms inside the square root. The expression inside the square root is $6^2+8^2$, which can be calculated as follows:

$6^2 = 36$

$8^2 = 64$

Adding these two values together, we get:

$36 + 64 = 100$

So, the expression inside the square root is equal to 100.

Simplifying the Square Root

Now that we have the value inside the square root, we can simplify the expression. The square root of 100 is 10, since $10^2 = 100$. Therefore, the expression $\sqrt{6^2+8^2}$ simplifies to 10.

Evaluating the Fraction

The next step is to evaluate the fraction $\frac{3}{8}$. This fraction is already simplified, so we can move on to the next step.

Evaluating the Second Term

The second term in the expression is $(1.5)^2$. To evaluate this term, we need to square 1.5. Squaring a number means multiplying it by itself. Therefore, $(1.5)^2 = 1.5 \times 1.5 = 2.25$.

Combining the Terms

Now that we have evaluated both terms, we can combine them to simplify the expression. The expression becomes:

$\frac{3}{8} \times 10 + 2.25$

Simplifying the Expression

To simplify the expression, we need to multiply the fraction by the value inside the square root. Multiplying a fraction by a number means multiplying the numerator by the number and keeping the denominator the same. Therefore, $\frac{3}{8} \times 10 = \frac{3 \times 10}{8} = \frac{30}{8}$.

Simplifying the Fraction

The fraction $\frac{30}{8}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2. Therefore, $\frac{30}{8} = \frac{15}{4}$.

Combining the Terms

Now that we have simplified the fraction, we can combine the terms to get the final answer. The expression becomes:

$\frac{15}{4} + 2.25$

Evaluating the Expression

To evaluate the expression, we need to add the two terms together. Adding a fraction and a decimal number means converting the fraction to a decimal number and then adding the two numbers. Therefore, $\frac{15}{4} = 3.75$, and adding 2.25 to it gives us:

$3.75 + 2.25 = 6$

Conclusion

In this article, we have evaluated the expression $\frac{3}{8} \sqrt{6^2+8^2} + (1.5)^2$ using a step-by-step approach. We started by simplifying the expression inside the square root, then evaluated the fraction and the second term, and finally combined the terms to get the final answer. The final answer is 6.

Frequently Asked Questions

• Q: What is the value of $\sqrt{6^2+8^2}$? A: The value of $\sqrt{6^2+8^2}$ is 10.
• Q: What is the value of $(1.5)^2$? A: The value of $(1.5)^2$ is 2.25.
• Q: What is the final answer to the expression $\frac{3}{8} \sqrt{6^2+8^2} + (1.5)^2$? A: The final answer to the expression $\frac{3}{8} \sqrt{6^2+8^2} + (1.5)^2$ is 6.

Final Answer

The final answer to the expression $\frac{3}{8} \sqrt{6^2+8^2} + (1.5)^2$ is 6.

Introduction

In our previous article, we evaluated the expression $\frac{3}{8} \sqrt{6^2+8^2} + (1.5)^2$ using a step-by-step approach. We received many questions from readers who were struggling to understand the concepts and techniques used in the solution. In this article, we will answer some of the most frequently asked questions related to evaluating expressions with square roots and fractions.

Q&A

Q: What is the difference between a square root and a fraction?

A: A square root is a mathematical operation that finds the number that, when multiplied by itself, gives a specified value. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16. A fraction, on the other hand, is a way of expressing a part of a whole as a ratio of two numbers. For example, the fraction 3/4 represents three-fourths of a whole.

Q: How do I simplify a square root expression?

A: To simplify a square root expression, you need to find the largest perfect square that divides the number inside the square root. For example, the square root of 18 can be simplified as follows:

$\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}$

Q: How do I add a fraction and a decimal number?

A: To add a fraction and a decimal number, you need to convert the fraction to a decimal number and then add the two numbers. For example, to add 3/4 and 2.5, you can convert the fraction to a decimal number as follows:

$3/4 = 0.75$

Then, you can add the two numbers:

$0.75 + 2.5 = 3.25$

Q: What is the order of operations when evaluating an expression with square roots and fractions?

A: The order of operations when evaluating an expression with square roots and fractions is as follows:

1. Evaluate any expressions inside parentheses.
2. Evaluate any exponents (such as squaring or cubing).
3. Evaluate any multiplication and division operations from left to right.
4. Evaluate any addition and subtraction operations from left to right.

Q: How do I simplify a fraction with a square root in the numerator or denominator?

A: To simplify a fraction with a square root in the numerator or denominator, you need to find the largest perfect square that divides the numerator or denominator. For example, the fraction 3/√2 can be simplified as follows:

$3/√2 = 3√2/2$

Q: What is the difference between a rational number and an irrational number?

A: A rational number is a number that can be expressed as a ratio of two integers. For example, the number 3/4 is a rational number. An irrational number, on the other hand, is a number that cannot be expressed as a ratio of two integers. For example, the square root of 2 is an irrational number.

Q: How do I determine if a number is rational or irrational?

A: To determine if a number is rational or irrational, you need to check if it can be expressed as a ratio of two integers. If it can be expressed as a ratio of two integers, it is a rational number. If it cannot be expressed as a ratio of two integers, it is an irrational number.

Conclusion

In this article, we have answered some of the most frequently asked questions related to evaluating expressions with square roots and fractions. We hope that this article has provided you with a better understanding of the concepts and techniques used in evaluating such expressions.

Frequently Asked Questions

• Q: What is the difference between a square root and a fraction? A: A square root is a mathematical operation that finds the number that, when multiplied by itself, gives a specified value. A fraction, on the other hand, is a way of expressing a part of a whole as a ratio of two numbers.
• Q: How do I simplify a square root expression? A: To simplify a square root expression, you need to find the largest perfect square that divides the number inside the square root.
• Q: How do I add a fraction and a decimal number? A: To add a fraction and a decimal number, you need to convert the fraction to a decimal number and then add the two numbers.
• Q: What is the order of operations when evaluating an expression with square roots and fractions? A: The order of operations when evaluating an expression with square roots and fractions is as follows: 1. Evaluate any expressions inside parentheses. 2. Evaluate any exponents (such as squaring or cubing). 3. Evaluate any multiplication and division operations from left to right. 4. Evaluate any addition and subtraction operations from left to right.

Final Answer

The final answer to the expression $\frac{3}{8} \sqrt{6^2+8^2} + (1.5)^2$ is 6.