Nice Square Root Math Simplification | Find the Value of X

Nice Square Root Math Simplification | Find the Value of X

[@tehatte]

Convert the exponents to fractions on both sides. It’s quicker. X^3/4=3^3/2 Now power 4/3 on both sides X=(3^3/2)^4/3=3^2=9

[@Nikioko]

x√x = (3√3)² = 9√9. x = 9 That is the obvious real solution. But there are two complex ones as well: x√x = (3√3)² √x³ = (3√3)² x³ = (3√3)⁴ x³ = 729 x³ − 729 = 0 Since we already know that x − 9 = 0 for x₁ = 9 is a solution, we can factor that out: (x − 9) (x² + 9x + 81) = 0 According to the rule of the zero product, the equation is zero when one of the factors is zero. We already know that for x₁ = 9, the first factor gets zero. So, let's solve for the second: x² + 9x + 81 = 0 x₂,₃ = −9/2 ± √(81/4 − 81) = −9/2 ± √(−243/4) = −9/2 ± i√(243/4) = −9/2 ± i√243 / 2 = −9/2 ± 9i√3 / 2 x₂ = (−9 − 9i√3)/2 x₃ = (−9 + 9i√3)/2 And now we have our three solutions.

[@siyeducation]

Woww... that was actually sooo long to get a simple solution. X^3 = 27*9 = 729 --> x = +9 for real values (non imaginary).

[@davidderby6951]

What do the hats or carpors on numbers mean?

[@SuperAnangs]

In 7 sec, I have solved x=9

[@ta192utube]

Simplification? Really?

[@user-dq3uh6ee5w]

9.

[@guitC]

you can find solution earlier, if x^3=9^3, then simply take root 3 from.both sides and you.will have solution that x = 9😊

[@sylvesterogbolu-otutu1498]

There is only 1 solution, really: x = 9 € N. All your clever manipulations are not necessary. Instead of writing x^3 = 9^3, you could have equally written x^3 = 3^6, and taking the cube root of both sides would have given x = 3^2 = 9. Going the way of x^3 - 9^3 = 0 led you to complex and irrational roots that are not required, since doing it differently (x^3 = 3^6) would have resulted in only x = 9.

[@Nikioko]

The problem is that x³ = 27 is a third degree polynomial. And a third degree polynomial of course has three solutions. One of them is x = 9. But there are two more: x³ = 729 x³ − 729 = 0 Since you already know that x − 9 = 0 for x₁ = 9 is a solution, you can factor that out: (x − 9) (x² + 9x + 81) = 0 According to the rule of the zero product, the equation is zero when one of the factors is zero. We already know that for x₁ = 9, the first factor gets zero. So, lets solve for the second: x² + 9x + 81 = 0 x₂,₃ = −9/2 ± √(81/4 − 81) = −9/2 ± √(−243/4) The root of a negative radicand is not defined for real numbers. However, for complex numbers, the definition i² = −1 means that √(−x) = i√x. Or in this case: x₂,₃ = −9/2 ± √(−243/4) = −9/2 ± i√(243/4) = −9/2 ± i√243 / 2 x₂ = (−9 − i√243)/2 x₃ = (−9 + i√243)/2 And now we have our three solutions. Ah, you can still replace √243 by 9√9, but that doesn't change the result.

[@johngeverett]

I cannot see that the last 2 'solutions' are actually solutions.

[@Nikioko]

They are complex solutions. A third degree polynomial always has three solutions.

[@ericzacher509]

True that but they are not really of any use (at least I think so)

[@Nikioko]

@@ericzacher509 In this equation or in general? In this equation, they fulfil the rule that every polynomial function has as many solutions as the highest exponent. In real life, complex numbers have a role in physics, like alternating current or the Schrödinger Equation.

[@ericzacher509]

Yeah you are right with that. I meant that you cant really the non-real solutions im any practical way (besides in higher maths and the likes)

[@Nikioko]

@@ericzacher509 As I said, complex numbers are important in quantum physics and electrical engineering.

[@apulse2apluskmkasim769]

X³=27 X = 3, or -3

[@Nikioko]

A third degree polynomial has three solutions. So, your answer is incomplete. And −3 is wrong.

[@guitC]

not 27 but 81

[@Igor28236]

задача решается в уме в два действия.

[@vics8873]

Silly

[@Nikioko]

What is silly?

[@vics8873]

@@Nikioko too easy...

[@Nikioko]

@@vics8873 Too easy to get all three solutions?

[@vics8873]

@@Nikioko for most of us who went to high school...

[@Nikioko]

@@vics8873 You went to high school? In the US? And there you had complex numbers?

[@yanfisher2639]

Too much unnecessary writing, many nonsense