Video: Expanding Numerical Expressions Involving Square Roots

Expand (√(6) − 7)².

02:26

Video Transcript

Expand root six minus seven all squared.

To expand this parenthesis, what we’re gonna do is first rewrite it. So we’ve actually got root six minus seven multiplied by root six minus seven. Then to expand them, what we actually do, we do it the same way that we do an algebraic expansion. It doesn’t matter to include surds. We carry out in the same way.

So we’re gonna start by multiplying the first term in each of our parentheses. So we got root six multiplied by root six. And this is gonna give us root 36 because we have the rule that says that root 𝑎 multiplied by root 𝑏 is equal to root 𝑎𝑏.

And then, we’re gonna multiply the first term in the first parenthesis by the last term in the second parenthesis. So we’ve got root six multiplied by negative seven, which will give us negative seven root six. Then next, we have the second term in the first parenthesis multiplied by the first term in the second parenthesis, which is negative seven multiplied by root six. So again, we get negative seven root six.

And then finally, we multiply the last two terms in our parentheses. So we get negative seven multiplied by negative seven, which gives us positive 49. And we get positive 49 because a negative multiplied by a negative is a positive.

So now our next stage is to actually simplify our expression. So first of all, we have root 36, which is just gonna give us six. So it’s actually worth noting at this point that we could have actually arrived at this six after the first stage because root 𝑎 multiplied by root 𝑎 is just equal to 𝑎. So if we had root six multiplied by root six, that’s just equal to six. So root 10 multiplied by root 10 is just equal to 10. But what I wanted to do is actually to show you where this came from.

Then next, we have minus 14 root six. And we have that because we had negative seven root six minus seven root six. So we have minus 14 root six and then we have plus 49 on the end. Okay, great, one more step so that we can fully simplify. We have six plus 49, which is 55, and then minus 14 root six.

So therefore, we can say that expanded and fully simplified, root six minus seven all squared is equal to 55 minus 14 root six.

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