Video: Operations on Square Roots in Word Problems

Given that the dimensions of a rectangle are (12 + √2) cm and (12 − √2) cm, find its area.

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Video Transcript

Given that the dimensions of a rectangle are 12 plus root two centimeters and 12 minus root two centimeters, find its area.

So in this question, we’re actually looking to find the area of a rectangle. So I’ve drawn a little sketch here of our rectangle. And I’ve put on our different values that we have for its dimensions. So we have 12 plus root two and 12 minus root two.

Well, to find the area of a rectangle, we know what we need to do is multiply the length by the width. So in our rectangle, what that means is, we’re actually gonna multiply 12 plus root two by 12 minus root two.

Okay, so now let’s get on and actually multiply them. Well, to multiply them, because they are both expressions in parentheses, we’re actually going to expand the parentheses. So first of all, we’re gonna multiply the first term in each of our parentheses. So we have 12 multiplied by 12, which gives us 144.

Next, we’re gonna multiply our 12 by negative root two, which gives us negative 12 root two. Then next, we’re gonna multiply the second term in our first parentheses by the first term in our second parentheses. So we’re gonna get root two multiplied by 12, which will give us plus 12 root two. And then finally, we’re gonna have root two multiplied by negative root two.

So then we’re gonna get minus root two root two. And we get a negative because if you multiply a positive and a negative, you end up with a negative. Okay, great! So let’s simplify this. So we’re gonna have that 𝐴, our area, is equal to 144. And then we’ve got minus 12 root two plus 12 root two. So they’re gonna cancel each other out. And that equals zero. And then it’s minus. And we’ve got root two multiplied by root two. And here we use the rule that root 𝑎 multiplied by root 𝑎, or root 𝑎 squared, is just equal to 𝑎.

So in this case, we’re gonna have root two multiplied by root two just gives us two. So we’re left with 𝐴 is equal to 144 minus two. So therefore, we can say that given the dimensions of our rectangle are 12 plus root two centimeters and 12 minus root two centimeters, then our area is gonna be equal to 142 centimeters squared.

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