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What is the difference in surface area between cube with an edge length of 16 inches and a cube with an edge length of five inches?
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Before we start, a few things we should know.
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We’re dealing with surface area.
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Surface area of a cube is the area of one side, so side squared, multiplied by six because a cube is made of six equal faces.
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One of our cubes has a side length of 16 inches.
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That’s the larger square.
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So we’ll say the surface area of the larger equals six times 16 squared.
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Now the smaller, surface area of the smaller is equal to six times five squared.
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We want to take the difference of these two.
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Six times 16 squared minus six times five squared.
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You notice that both of these terms are being multiplied by six.
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So we can undistribute that six.
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Our new equation would say six times 16 squared minus five squared.
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Here’s where a big temptation will come in.
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You might want to do this.
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You might want to subtract five from 16 and then square it.
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However, this is not true.
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We do not distribute exponents over addition and subtraction.
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And you’ll see why.
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16 squared equals 256.
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Five squared equals 25.
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We now have six times 256 minus 25.
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256 minus 25 equals 231.
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Six times 231equals 1386.
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Remember we’re dealing with surface area.
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And that means our unit is inches squared.
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Let’s go back to this distribution of exponents idea and see why it doesn’t work.
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If you had begun by subtracting five from 16, your new statement would say 11 squared times six.
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11 squared equals 121.
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Multiply that value by six and you get 726.
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It’s not the same thing.
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I wanna show you what this looks like with an image.
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Our small square is five by five.
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Our large square is 16.
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That means we have a partial length of 11 on two sides and then a length of 16 on the other two sides.
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The difference in area is how much bigger the blue square is than the yellow square.
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If we break that difference up into these three spaces, we have a square that measures 11 by 11, a rectangle that measures 11 by five, and another square that measures 11 by five.
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See this difference of 11 squared?
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It only accounts for this tiny corner of difference.
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We have this 11 squared equal to 121, 11 times five equals 55, and 11 times five equals 55.
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If you add up these three values, it equals 231.
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And that’s why we have to take the full value of 16 squared, 256, the full value of 16 squared and then subtract the corner, 25, from that value.
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After we find the surface area difference, and the final answer is 1386 inches squared.