Video: Evaluating Algebraic Expressions Involving Irrational numbers

Given that 𝑥 = √(3 + √7), find the value of 𝑥⁴ + 4𝑥² + 4.

03:36

Video Transcript

Given that 𝑥 equals square root of three plus root seven, find the value of 𝑥 to the power of four plus four 𝑥 squared plus four.

In order to solve this problem, what I’m going to need to do is substitute in 𝑥 equals the square root of three plus root seven into our expression, which is 𝑥 to the power of four plus four 𝑥 squared plus four. When we substitute in our value for 𝑥, what we get is the square root of three plus root seven all to the power of four plus four multiplied by the square root of three plus root seven all squared plus four. So it’s now that we’re gonna use some exponent rules to help us take this even further.

The first rule we have is that the square root of 𝑎 can be written as 𝑎 to the power of a half. So therefore, using this rule, we can rewrite our expression as three plus root seven all to the power of a half and then this all to the power of four plus four multiplied by, then we’ve got three plus root seven to the power of a half, then this is all to the power of two all squared plus four. The next rule that we have, and it’s going to help us to simplify further, is the one that tells us that if we have 𝑥 to the power of 𝑎 and then this is to the power of 𝑏 it’s equal to 𝑥 to the power of 𝑎 multiplied by 𝑏.

So then if we take a look back at our expression, we can see that for the first term we’re gonna multiply the power of a half by four, which will leave us with three plus root seven all squared, then plus four multiplied by three plus root seven root. And we’ve just got three plus root seven because we had three plus root seven to the power of a half, and this all to the power of two. Well if you multiply a half by two, you get one. And then finally, we’ve got our add four on the end.

So now what we need to do is expand our parentheses. Well if we expand three plus root seven all squared, we’re gonna have three plus root seven multiplied by three plus root seven. And we start with three multiplied by three, which is nine, then plus three root seven because we’ve got three multiplied by root seven, then plus another three root seven cause again we’ve got positive root seven multiplied by three. And then finally we have add seven, and that’s cause you’ve positive root seven multiplied by positive root seven. And that’s because we have a rule that tells us that if we have root 𝑎 multiplied by root 𝑎, it just gives us 𝑎. And this is because if we had root seven multiplied by root seven, we’d have root 49. And root 49 is equal to seven.

And therefore, if we collect like terms here, we’re gonna get 16 plus six root seven. And then we’ll have plus 12 cause we’ve got four multiplied by three in the second parenthesis, and then plus four root seven because you’ve got four multiplied by root seven, and then finally add four because we still got the four on the end. Okay, great! So we’re now at a stage where all we need to do is collect like terms and we’ll be fully simplified. So first of all we’ve got 16 add 12, which is 28, add four, which is 32. And then we have positive six root seven plus four root seven, which gives us 10 root seven. So therefore, we can say that given that 𝑥 equals the square root of three plus root seven, then the value of 𝑥 to the power of four plus four 𝑥 squared plus four is equal to 32 plus 10 root seven, or it can be written as 10 root seven plus 32.

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