Question

the area of a square room can be expressed as 196x¹²y¹⁰z⁶ square feet. hajime is installing a wallpaper border along the perimeter of the room. what is the perimeter?
○ 52x⁶y⁵z³ feet
○ 52x¹⁰y⁸z⁴ feet
○ 56x¹⁰y⁸z⁴ feet
○ 56x⁶y⁵z³ feet

Explanation:

Step1: Recall square area formula

The area \( A \) of a square is given by \( A = s^2 \), where \( s \) is the side length. So, to find \( s \), we take the square root of the area.
Given \( A = 196x^{12}y^{10}z^6 \), then \( s=\sqrt{196x^{12}y^{10}z^6} \).
Simplify each part: \( \sqrt{196} = 14 \), \( \sqrt{x^{12}} = x^{6} \) (since \( (x^6)^2 = x^{12} \)), \( \sqrt{y^{10}} = y^{5} \) (since \( (y^5)^2 = y^{10} \)), \( \sqrt{z^6} = z^{3} \) (since \( (z^3)^2 = z^6 \)). So, \( s = 14x^6y^5z^3 \).

Step2: Recall square perimeter formula

The perimeter \( P \) of a square is \( P = 4s \).
Substitute \( s = 14x^6y^5z^3 \) into the formula: \( P = 4\times14x^6y^5z^3 \).
Calculate \( 4\times14 = 56 \), so \( P = 56x^6y^5z^3 \) feet.

Answer:

\( 56x^{6}y^{5}z^{3} \) feet (matching the last option, assuming the typo in the option is a misprint and should be \( 56x^6y^5z^3 \) instead of \( 56x^6y^5z^3 \) with possible typo in variable name, but the exponent and coefficient match the calculation)