Axe and Why

Algebra Level 3

If $xy=6$ and $x^2 y + y^2 x + x+ y=63$ . Let $S$ denote the sum of digits of the number $(x^4+y^4)(x^3+y^3)(x^2+y^2)(x+y)$ .

Find the value of $\sqrt[3]{\sqrt S + \frac S4 + \frac S3 }$ .

$Note$ :You can use calculator at final calculation.

The answer is 3.

2 solutions

Joe Potillor
Nov 4, 2016

I am typing this solution from my phone, so bear with any format errors. For this problem, not so bad, we need to use the binomial expansions to make some of the calculations easier...

(a + b)^n, for n = 1, 2, 3, 4

If we use the given xy = 6, and we plug it into the 2nd equation x^2(y) + y^2(x) + X + y = 63, we get (xy)x + (xy)y + X + y = 63

6x + 6y + x + y = 7x + 7y = 63 = 7 (x + y) = 63 = (x + y) = 9

From here

x + y

x^2 +y^2 = (x + y)^2-(2xy)

x^3 +y^3 = (x +y)^3-3xy(x +y)

x^4 + y^4 = (x + y)^4 -6 (xy)^2 - 4xy ((x + y)^2 - 2 (xy))

Plugging in xy and x + y, into the above expressions you will get

9 69 567 3033

Multiplying the above results you will get

1 067 940 531

Since s is defined as the sum of these numbers, we add them together to get 36

Lastly, we plug into equation we want to solve

Square root of 36 is 6, 36/4 = 9, 36/3 = 12

6 +9 + 12 = 27

Cube root of 27 is 3 and that is the answer.

Akash Mehta
Jul 11, 2015

without calculator isn't this problem too lenghty it took my 20 minutes to solve isn't there any short trick @Akshat Sharda

sakshi rathore - 5 years, 11 months ago

By the way it won't take much time .Also I got a solution to your problem-“ $Now$ $read$ $the$ $question$ $again$ ” :-)

Akshat Sharda - 5 years, 10 months ago

Axe and Why

The answer is 3.

2 solutions

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