Evaluate The Expression:${ X = 4 \pm \sqrt{\frac{8}{4}} }$

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Introduction

In mathematics, expressions involving square roots and fractions are common and can be challenging to evaluate. The given expression, $x = 4 \pm \sqrt{\frac{8}{4}}$, is a classic example of such an expression. In this article, we will evaluate the expression and provide a step-by-step solution to understand the concept better.

Understanding the Expression

The given expression is $x = 4 \pm \sqrt{\frac{8}{4}}$. To evaluate this expression, we need to follow the order of operations (PEMDAS):

1. Evaluate the fraction inside the square root.
2. Simplify the square root.
3. Add or subtract the simplified square root from 4.

Evaluating the Fraction Inside the Square Root

The fraction inside the square root is $\frac{8}{4}$. To simplify this fraction, we can divide the numerator (8) by the denominator (4):

$$\frac{8}{4} = 2$$

Simplifying the Square Root

Now that we have simplified the fraction, we can simplify the square root:

$$\sqrt{\frac{8}{4}} = \sqrt{2}$$

Evaluating the Expression

Now that we have simplified the square root, we can evaluate the expression:

$$x = 4 \pm \sqrt{\frac{8}{4}}$$

$$x = 4 \pm \sqrt{2}$$

Understanding the Plus-Minus Sign

The plus-minus sign ($\pm$) indicates that there are two possible solutions to the expression. The two solutions are:

$$x = 4 + \sqrt{2}$$

$$x = 4 - \sqrt{2}$$

Evaluating the Two Solutions

To evaluate the two solutions, we can simplify each expression:

$$x = 4 + \sqrt{2}$$

$$x = 4 + 1.414$$

$$x = 5.414$$

$$x = 4 - \sqrt{2}$$

$$x = 4 - 1.414$$

$$x = 2.586$$

Conclusion

In this article, we evaluated the expression $x = 4 \pm \sqrt{\frac{8}{4}}$ and provided a step-by-step solution to understand the concept better. We simplified the fraction inside the square root, simplified the square root, and evaluated the two possible solutions. The two solutions are $x = 4 + \sqrt{2}$ and $x = 4 - \sqrt{2}$.

Final Answer

The final answer to the expression $x = 4 \pm \sqrt{\frac{8}{4}}$ is:

$$x = 4 + \sqrt{2}$$

$$x = 4 - \sqrt{2}$$

Frequently Asked Questions

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS stands for:

• Parentheses
• Exponents
• Multiplication and Division
• Addition and Subtraction

Q: How do I simplify a fraction?

A: To simplify a fraction, we can divide the numerator by the denominator.

Q: How do I simplify a square root?

A: To simplify a square root, we can look for perfect squares that divide the number inside the square root.

Q: What is the plus-minus sign?

A: The plus-minus sign ($\pm$) indicates that there are two possible solutions to an expression.

References

• [1] Khan Academy. (n.d.). Order of Operations. Retrieved from https://www.khanacademy.org/math/algebra/x2f0f7d
• [2] Mathway. (n.d.). Simplifying Fractions. Retrieved from https://www.mathway.com/subjects/simplifying-fractions
• [3] Purplemath. (n.d.). Simplifying Square Roots. Retrieved from https://www.purplemath.com/modules/sqrt.htm
• [4] Wolfram Alpha. (n.d.). Plus-Minus Sign. Retrieved from https://www.wolframalpha.com/input/?i=plus-minus+sign
  

Introduction

In our previous article, we evaluated the expression $x = 4 \pm \sqrt{\frac{8}{4}}$ and provided a step-by-step solution to understand the concept better. In this article, we will answer some frequently asked questions related to evaluating expressions with square roots and fractions.

Q&A

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS stands for:

• Parentheses
• Exponents
• Multiplication and Division
• Addition and Subtraction

Q: How do I simplify a fraction?

A: To simplify a fraction, we can divide the numerator by the denominator. For example, to simplify the fraction $\frac{8}{4}$, we can divide the numerator (8) by the denominator (4):

$$\frac{8}{4} = 2$$

Q: How do I simplify a square root?

A: To simplify a square root, we can look for perfect squares that divide the number inside the square root. For example, to simplify the square root $\sqrt{16}$, we can look for the perfect square that divides 16:

$$\sqrt{16} = \sqrt{4 \times 4} = 4$$

Q: What is the plus-minus sign?

A: The plus-minus sign ($\pm$) indicates that there are two possible solutions to an expression. For example, in the expression $x = 4 \pm \sqrt{2}$, there are two possible solutions:

$$x = 4 + \sqrt{2}$$

$$x = 4 - \sqrt{2}$$

Q: How do I evaluate an expression with multiple operations?

A: To evaluate an expression with multiple operations, we can follow the order of operations (PEMDAS). For example, to evaluate the expression $x = 3 \times 2 + 4 - 1$, we can follow the order of operations:

1. Multiply 3 and 2:

$$3 \times 2 = 6$$

2. Add 4 to 6:

$$6 + 4 = 10$$

3. Subtract 1 from 10:

$$10 - 1 = 9$$

Q: How do I handle negative numbers in expressions with square roots?

A: When we have a negative number inside a square root, we can simplify it by multiplying the number by -1. For example, to simplify the square root $\sqrt{-16}$, we can multiply the number by -1:

$$\sqrt{-16} = \sqrt{-1 \times 16} = \sqrt{-1} \times \sqrt{16} = 4i$$

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 the ratio of two integers, i.e., a fraction. An irrational number is a number that cannot be expressed as the ratio of two integers. For example, the number $\sqrt{2}$ is an irrational number because it cannot be expressed as the ratio of two integers.

Conclusion

In this article, we answered some frequently asked questions related to evaluating expressions with square roots and fractions. We covered topics such as the order of operations, simplifying fractions and square roots, and handling negative numbers in expressions with square roots.

Final Answer

The final answer to the expression $x = 4 \pm \sqrt{\frac{8}{4}}$ is:

$$x = 4 + \sqrt{2}$$

$$x = 4 - \sqrt{2}$$

Frequently Asked Questions

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when we have multiple operations in an expression.

Q: How do I simplify a fraction?

A: To simplify a fraction, we can divide the numerator by the denominator.

Q: How do I simplify a square root?

A: To simplify a square root, we can look for perfect squares that divide the number inside the square root.

Q: What is the plus-minus sign?

A: The plus-minus sign ($\pm$) indicates that there are two possible solutions to an expression.

References

• [1] Khan Academy. (n.d.). Order of Operations. Retrieved from https://www.khanacademy.org/math/algebra/x2f0f7d
• [2] Mathway. (n.d.). Simplifying Fractions. Retrieved from https://www.mathway.com/subjects/simplifying-fractions
• [3] Purplemath. (n.d.). Simplifying Square Roots. Retrieved from https://www.purplemath.com/modules/sqrt.htm
• [4] Wolfram Alpha. (n.d.). Plus-Minus Sign. Retrieved from https://www.wolframalpha.com/input/?i=plus-minus+sign