### Video Transcript

Given that ๐ด๐ท equals ๐ฅ to the fourth ๐ฆ cubed squared centimeters and ๐ถ๐ต equals six ๐ฅ๐ฆ to the fourth squared centimeters, determine in its simplest form the algebraic expression that represents the area of the triangle ๐ด๐ต๐ถ.

So it seems like we have a lot going on here; letโs break it down. Weโre given the length of ๐ด๐ท, which is the height of our triangle. They tell us that that equals ๐ฅ to the fourth ๐ฆ cubed squared. And then weโre given the distance of ๐ถ๐ต, which is the base of our triangle, a height of ๐ฅ to the fourth ๐ฆ cubed squared and a base of six ๐ฅ๐ฆ to the fourth squared.

Before we find the area, letโs 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 weโre taking the power of a power, we multiply the exponents. And thatโs what weโll do here; weโll multiply four times two to find the power of our ๐ฅ; and weโll multiply three times two to find the power of our ๐ฆ.

So simplified, our height can be written as ๐ฅ to the eighth power times ๐ฆ to the sixth power. Weโll also need to distribute our powers to simplify the base. In this case, weโll get six squared times ๐ฅ squared, and then weโll multiply four times two to find the exponent of ๐ฆ. And the simplified form of our base becomes 36๐ฅ squared ๐ฆ 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 ๐ฅ to the eighth ๐ฆ to the sixth times 36๐ฅ squared ๐ฆ to the eighth.

What will do now is combine our like terms: one half times 36 equals 18; ๐ฅ to the eighth power times ๐ฅ squared equals ๐ฅ to the tenth power โ here we add our exponents because weโre multiplying two powers with the same base โ and finally ๐ฆ to the sixth power times ๐ฆ to the eighth power.

This is ๐ฆ 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 18๐ฅ to the tenth ๐ฆ to the 14th centimeters squared.