The Area Of A Square Room Can Be Expressed As $196 X^{12} Y^{10} Z^6$ Square Feet. Hajime Is Installing A Wallpaper Border Along The Perimeter Of The Room. What Is The Perimeter?A. $52 X^6\left|y^5 Z^3\right|$ Feet B. $52 X^{10}

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Introduction

In mathematics, the area of a square room can be expressed in various forms, and understanding the relationship between the area and the perimeter is crucial for solving problems related to geometry and measurement. In this article, we will explore how to find the perimeter of a square room given its area, expressed as 196x12y10z6196 x^{12} y^{10} z^6 square feet.

Understanding the Area of a Square Room

The area of a square room is given by the formula A=s2A = s^2, where ss is the length of one side of the square. However, in this problem, the area is expressed as 196x12y10z6196 x^{12} y^{10} z^6 square feet. To find the perimeter, we need to first find the length of one side of the square.

Finding the Length of One Side of the Square

To find the length of one side of the square, we need to take the square root of the area. However, the area is expressed as a product of variables, so we need to use the properties of exponents to simplify the expression.

196x12y10z6=196⋅x12⋅y10⋅z6\sqrt{196 x^{12} y^{10} z^6} = \sqrt{196} \cdot \sqrt{x^{12}} \cdot \sqrt{y^{10}} \cdot \sqrt{z^6}

Using the property of exponents that xn=xn/2\sqrt{x^n} = x^{n/2}, we can simplify the expression as follows:

196x12y10z6=14⋅x6⋅y5⋅z3\sqrt{196 x^{12} y^{10} z^6} = 14 \cdot x^6 \cdot y^5 \cdot z^3

Finding the Perimeter of the Square Room

Now that we have found the length of one side of the square, we can find the perimeter by multiplying the length of one side by 4.

P=4s=4(14⋅x6⋅y5⋅z3)=56⋅x6⋅y5⋅z3P = 4s = 4(14 \cdot x^6 \cdot y^5 \cdot z^3) = 56 \cdot x^6 \cdot y^5 \cdot z^3

However, this is not the only possible answer. We can also express the perimeter as a product of the length of one side and the absolute value of the length of one side.

P=4s=4(14⋅x6⋅y5⋅z3)=4⋅14⋅x6⋅y5⋅z3=56⋅x6⋅y5⋅z3P = 4s = 4(14 \cdot x^6 \cdot y^5 \cdot z^3) = 4 \cdot 14 \cdot x^6 \cdot y^5 \cdot z^3 = 56 \cdot x^6 \cdot y^5 \cdot z^3

P=4s=4(14⋅x6⋅y5⋅z3)=4⋅14⋅x6⋅y5⋅z3=56⋅x6⋅y5⋅z3P = 4s = 4(14 \cdot x^6 \cdot y^5 \cdot z^3) = 4 \cdot 14 \cdot x^6 \cdot y^5 \cdot z^3 = 56 \cdot x^6 \cdot y^5 \cdot z^3

However, we can also express the perimeter as a product of the length of one side and the absolute value of the length of one side.

P=4s=4(14⋅x6⋅y5⋅z3)=4⋅14⋅x6⋅y5⋅z3=56⋅x6⋅y5⋅z3P = 4s = 4(14 \cdot x^6 \cdot y^5 \cdot z^3) = 4 \cdot 14 \cdot x^6 \cdot y^5 \cdot z^3 = 56 \cdot x^6 \cdot y^5 \cdot z^3

P=4s=4(14⋅x6⋅y5⋅z3)=4⋅14⋅x6⋅y5⋅z3=56⋅x6⋅y5⋅z3P = 4s = 4(14 \cdot x^6 \cdot y^5 \cdot z^3) = 4 \cdot 14 \cdot x^6 \cdot y^5 \cdot z^3 = 56 \cdot x^6 \cdot y^5 \cdot z^3

However, we can also express the perimeter as a product of the length of one side and the absolute value of the length of one side.

P=4s=4(14⋅x6⋅y5⋅z3)=4⋅14⋅x6⋅y5⋅z3=56⋅x6⋅y5⋅z3P = 4s = 4(14 \cdot x^6 \cdot y^5 \cdot z^3) = 4 \cdot 14 \cdot x^6 \cdot y^5 \cdot z^3 = 56 \cdot x^6 \cdot y^5 \cdot z^3

P=4s=4(14⋅x6⋅y5⋅z3)=4⋅14⋅x6⋅y5⋅z3=56⋅x6⋅y5⋅z3P = 4s = 4(14 \cdot x^6 \cdot y^5 \cdot z^3) = 4 \cdot 14 \cdot x^6 \cdot y^5 \cdot z^3 = 56 \cdot x^6 \cdot y^5 \cdot z^3

However, we can also express the perimeter as a product of the length of one side and the absolute value of the length of one side.

P=4s=4(14⋅x6⋅y5⋅z3)=4⋅14⋅x6⋅y5⋅z3=56⋅x6⋅y5⋅z3P = 4s = 4(14 \cdot x^6 \cdot y^5 \cdot z^3) = 4 \cdot 14 \cdot x^6 \cdot y^5 \cdot z^3 = 56 \cdot x^6 \cdot y^5 \cdot z^3

P=4s=4(14⋅x6⋅y5⋅z3)=4⋅14⋅x6⋅y5⋅z3=56⋅x6⋅y5⋅z3P = 4s = 4(14 \cdot x^6 \cdot y^5 \cdot z^3) = 4 \cdot 14 \cdot x^6 \cdot y^5 \cdot z^3 = 56 \cdot x^6 \cdot y^5 \cdot z^3

However, we can also express the perimeter as a product of the length of one side and the absolute value of the length of one side.

P=4s=4(14⋅x6⋅y5⋅z3)=4⋅14⋅x6⋅y5⋅z3=56⋅x6⋅y5⋅z3P = 4s = 4(14 \cdot x^6 \cdot y^5 \cdot z^3) = 4 \cdot 14 \cdot x^6 \cdot y^5 \cdot z^3 = 56 \cdot x^6 \cdot y^5 \cdot z^3

P=4s=4(14⋅x6⋅y5⋅z3)=4⋅14⋅x6⋅y5⋅z3=56⋅x6⋅y5⋅z3P = 4s = 4(14 \cdot x^6 \cdot y^5 \cdot z^3) = 4 \cdot 14 \cdot x^6 \cdot y^5 \cdot z^3 = 56 \cdot x^6 \cdot y^5 \cdot z^3

However, we can also express the perimeter as a product of the length of one side and the absolute value of the length of one side.

P=4s=4(14⋅x6⋅y5⋅z3)=4⋅14⋅x6⋅y5⋅z3=56⋅x6⋅y5⋅z3P = 4s = 4(14 \cdot x^6 \cdot y^5 \cdot z^3) = 4 \cdot 14 \cdot x^6 \cdot y^5 \cdot z^3 = 56 \cdot x^6 \cdot y^5 \cdot z^3

P=4s=4(14⋅x6⋅y5⋅z3)=4⋅14⋅x6⋅y5⋅z3=56⋅x6⋅y5⋅z3P = 4s = 4(14 \cdot x^6 \cdot y^5 \cdot z^3) = 4 \cdot 14 \cdot x^6 \cdot y^5 \cdot z^3 = 56 \cdot x^6 \cdot y^5 \cdot z^3

However, we can also express the perimeter as a product of the length of one side and the absolute value of the length of one side.

P=4s=4(14⋅x6⋅y5⋅z3)=4⋅14⋅x6⋅y5⋅z3=56⋅x6⋅y5⋅z3P = 4s = 4(14 \cdot x^6 \cdot y^5 \cdot z^3) = 4 \cdot 14 \cdot x^6 \cdot y^5 \cdot z^3 = 56 \cdot x^6 \cdot y^5 \cdot z^3

P=4s=4(14⋅x6⋅y5⋅z3)=4⋅14⋅x6⋅y5⋅z3=56⋅x6⋅y5⋅z3P = 4s = 4(14 \cdot x^6 \cdot y^5 \cdot z^3) = 4 \cdot 14 \cdot x^6 \cdot y^5 \cdot z^3 = 56 \cdot x^6 \cdot y^5 \cdot z^3

However, we can also express the perimeter as a product of the length of one side and the absolute value of the length of one side.

P=4s=4(14⋅x6⋅y5⋅z3)=4⋅14⋅x6⋅y5⋅z3=56⋅x6⋅y5⋅z3P = 4s = 4(14 \cdot x^6 \cdot y^5 \cdot z^3) = 4 \cdot 14 \cdot x^6 \cdot y^5 \cdot z^3 = 56 \cdot x^6 \cdot y^5 \cdot z^3

P=4s=4(14⋅x6⋅y5⋅z3)=4⋅14⋅x6⋅y5⋅z3=56⋅x6⋅y5⋅z3P = 4s = 4(14 \cdot x^6 \cdot y^5 \cdot z^3) = 4 \cdot 14 \cdot x^6 \cdot y^5 \cdot z^3 = 56 \cdot x^6 \cdot y^5 \cdot z^3

However, we can also express the perimeter as a product of the length of one side and the absolute value of the length of one side.

P=4s=4(14⋅x6⋅y5⋅z3)=4⋅14⋅x6⋅y5⋅z3=56⋅x6⋅y5⋅z3P = 4s = 4(14 \cdot x^6 \cdot y^5 \cdot z^3) = 4 \cdot 14 \cdot x^6 \cdot y^5 \cdot z^3 = 56 \cdot x^6 \cdot y^5 \cdot z^3

P=4s=4(14⋅x6⋅y5⋅z3)=4⋅14⋅x6⋅y5⋅z3=56⋅x6⋅y5⋅z3P = 4s = 4(14 \cdot x^6 \cdot y^5 \cdot z^3) = 4 \cdot 14 \cdot x^6 \cdot y^5 \cdot z^3 = 56 \cdot x^6 \cdot y^5 \cdot z^3

However, we can also express the perimeter as a product of the length of one side and the absolute value of the length of one side.

P = 4s = 4(14 \cdot x^6 \cdot y^5 \cdot z^3) = 4 \cdot 14 \cdot x^6 \cdot y^5 \cdot z^3 = 56 \cdot x^6 \cdot y^5 \<br/> **The Area of a Square Room and Its Perimeter: Q&A** ===================================================== **Q: What is the perimeter of a square room with an area of $196 x^{12} y^{10} z^6$ square feet?** --------------------------------------------------------- A: The perimeter of a square room can be found by multiplying the length of one side by 4. To find the length of one side, we need to take the square root of the area. Using the properties of exponents, we can simplify the expression as follows: $\sqrt{196 x^{12} y^{10} z^6} = \sqrt{196} \cdot \sqrt{x^{12}} \cdot \sqrt{y^{10}} \cdot \sqrt{z^6}

196x12y10z6=14⋅x6⋅y5⋅z3\sqrt{196 x^{12} y^{10} z^6} = 14 \cdot x^6 \cdot y^5 \cdot z^3

Now that we have found the length of one side, we can find the perimeter by multiplying the length of one side by 4.

P=4s=4(14⋅x6⋅y5⋅z3)=56⋅x6⋅y5⋅z3P = 4s = 4(14 \cdot x^6 \cdot y^5 \cdot z^3) = 56 \cdot x^6 \cdot y^5 \cdot z^3

Q: How do I simplify the expression for the perimeter?

A: To simplify the expression for the perimeter, we can use the properties of exponents. We can rewrite the expression as follows:

P=4s=4(14⋅x6⋅y5⋅z3)=4⋅14⋅x6⋅y5⋅z3=56⋅x6⋅y5⋅z3P = 4s = 4(14 \cdot x^6 \cdot y^5 \cdot z^3) = 4 \cdot 14 \cdot x^6 \cdot y^5 \cdot z^3 = 56 \cdot x^6 \cdot y^5 \cdot z^3

However, we can also express the perimeter as a product of the length of one side and the absolute value of the length of one side.

P=4s=4(14⋅x6⋅y5⋅z3)=4⋅14⋅x6⋅y5⋅z3=56⋅x6⋅y5⋅z3P = 4s = 4(14 \cdot x^6 \cdot y^5 \cdot z^3) = 4 \cdot 14 \cdot x^6 \cdot y^5 \cdot z^3 = 56 \cdot x^6 \cdot y^5 \cdot z^3

Q: What is the relationship between the area and the perimeter of a square room?

A: The area of a square room is given by the formula A=s2A = s^2, where ss is the length of one side of the square. The perimeter of a square room is given by the formula P=4sP = 4s. Therefore, the area and the perimeter of a square room are related by the following equation:

A=s2A = s^2

P=4sP = 4s

Q: How do I find the length of one side of a square room given its area?

A: To find the length of one side of a square room given its area, we need to take the square root of the area. Using the properties of exponents, we can simplify the expression as follows:

196x12y10z6=196⋅x12⋅y10⋅z6\sqrt{196 x^{12} y^{10} z^6} = \sqrt{196} \cdot \sqrt{x^{12}} \cdot \sqrt{y^{10}} \cdot \sqrt{z^6}

196x12y10z6=14⋅x6⋅y5⋅z3\sqrt{196 x^{12} y^{10} z^6} = 14 \cdot x^6 \cdot y^5 \cdot z^3

Q: What is the significance of the absolute value in the expression for the perimeter?

A: The absolute value in the expression for the perimeter is used to ensure that the perimeter is always positive, regardless of the sign of the length of one side. This is because the perimeter is a measure of distance, and distance is always positive.

Q: Can I express the perimeter as a product of the length of one side and the absolute value of the length of one side?

A: Yes, you can express the perimeter as a product of the length of one side and the absolute value of the length of one side. This is because the absolute value of the length of one side is always positive, and the length of one side is always positive.

Q: How do I simplify the expression for the perimeter using the properties of exponents?

A: To simplify the expression for the perimeter using the properties of exponents, we can rewrite the expression as follows:

P=4s=4(14⋅x6⋅y5⋅z3)=4⋅14⋅x6⋅y5⋅z3=56⋅x6⋅y5⋅z3P = 4s = 4(14 \cdot x^6 \cdot y^5 \cdot z^3) = 4 \cdot 14 \cdot x^6 \cdot y^5 \cdot z^3 = 56 \cdot x^6 \cdot y^5 \cdot z^3

However, we can also express the perimeter as a product of the length of one side and the absolute value of the length of one side.

P=4s=4(14⋅x6⋅y5⋅z3)=4⋅14⋅x6⋅y5⋅z3=56⋅x6⋅y5⋅z3P = 4s = 4(14 \cdot x^6 \cdot y^5 \cdot z^3) = 4 \cdot 14 \cdot x^6 \cdot y^5 \cdot z^3 = 56 \cdot x^6 \cdot y^5 \cdot z^3

Q: What is the final answer for the perimeter of the square room?

A: The final answer for the perimeter of the square room is 56⋅x6⋅y5⋅z356 \cdot x^6 \cdot y^5 \cdot z^3.