Question Video: Finding the Area of a Triangle given Its Base and Height as Algebraic Terms | Nagwa Question Video: Finding the Area of a Triangle given Its Base and Height as Algebraic Terms | Nagwa

# Question Video: Finding the Area of a Triangle given Its Base and Height as Algebraic Terms Mathematics

Given that ๐ด๐ท = (๐ฅโด๐ฆยณ)ยฒ cm and ๐ถ๐ต = (6๐ฅ๐ฆโด)ยฒ cm, determine, in its simplest form, the algebraic expression that represents the area of triangle ๐ด๐ต๐ถ.

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### Video Transcript

Given that ๐ด๐ท equals ๐ฅ to the fourth ๐ฆ cubed squared centimeters and ๐ถ๐ต equals six ๐ฅ๐ฆ to the fourth squared centimeters, determine in its simplest form the algebraic expression that represents the area of the triangle ๐ด๐ต๐ถ.

So it seems like we have a lot going on here; letโs break it down. Weโre given the length of ๐ด๐ท, which is the height of our triangle. They tell us that that equals ๐ฅ to the fourth ๐ฆ cubed squared. And then weโre given the distance of ๐ถ๐ต, which is the base of our triangle, a height of ๐ฅ to the fourth ๐ฆ cubed squared and a base of six ๐ฅ๐ฆ to the fourth squared.

Before we find the area, letโ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 weโre taking the power of a power, we multiply the exponents. And thatโs what weโll do here; weโll multiply four times two to find the power of our ๐ฅ; and weโll multiply three times two to find the power of our ๐ฆ.

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

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

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

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