Given that 𝐴𝐷 equals 𝑥 to the fourth 𝑦 cubed squared centimeters and 𝐶𝐵 equals six 𝑥𝑦 to the fourth squared centimeters, determine in its simplest form the algebraic expression that represents the area of the triangle 𝐴𝐵𝐶.
So it seems like we have a lot going on here; let’s break it down. We’re given the length of 𝐴𝐷, which is the height of our triangle. They tell us that that equals 𝑥 to the fourth 𝑦 cubed squared. And then we’re given the distance of 𝐶𝐵, which is the base of our triangle, a height of 𝑥 to the fourth 𝑦 cubed squared and a base of six 𝑥𝑦 to the fourth squared.
Before we find the area, let’s simplify both the height and the base. To do that, I need to distribute the exponents, the powers, to the powers. And we remember our power to a power rule that says if we’re taking the power of a power, we multiply the exponents. And that’s what we’ll do here; we’ll multiply four times two to find the power of our 𝑥; and we’ll multiply three times two to find the power of our 𝑦.
So simplified, our height can be written as 𝑥 to the eighth power times 𝑦 to the sixth power. We’ll also need to distribute our powers to simplify the base. In this case, we’ll get six squared times 𝑥 squared, and then we’ll multiply four times two to find the exponent of 𝑦. And the simplified form of our base becomes 36𝑥 squared 𝑦 to the eighth power.
And we wanna find the area of this triangle. The formula for finding the area of a triangle equals one half height times the base. Our area will be one half times 𝑥 to the eighth 𝑦 to the sixth times 36𝑥 squared 𝑦 to the eighth.
What will do now is combine our like terms: one half times 36 equals 18; 𝑥 to the eighth power times 𝑥 squared equals 𝑥 to the tenth power — here we add our exponents because we’re multiplying two powers with the same base — and finally 𝑦 to the sixth power times 𝑦 to the eighth power.
This is 𝑦 to the 14th power. When we multiply a power times a power and they have the same base, we add the exponents. So we added six plus eight to equal 14. The area of this triangle would be equal to 18𝑥 to the tenth 𝑦 to the 14th centimeters squared.