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A negative real number −x has no real-valued square roots, but when x is treated as a complex number it has two imaginary square roots, + and , where i is the imaginary unit. In general, any non-zero complex number has n distinct complex-valued n th roots, equally distributed around a complex circle of constant absolute value .
For any integer n, the last decimal digit of n 5 is the same as the last (decimal) digit of n, i.e. ()By the Abel–Ruffini theorem, there is no general algebraic formula (formula expressed in terms of radical expressions) for the solution of polynomial equations containing a fifth power of the unknown as their highest power.
The allometric equation can also be acquired as a solution of the differential equation d y y = a d x x . {\displaystyle {\frac {dy}{y}}=a{\frac {dx}{x}}.} Allometry often studies shape differences in terms of ratios of the objects' dimensions.
For solving the cubic equation x 3 + m 2 x = n where n > 0, Omar Khayyám constructed the parabola y = x 2 /m, the circle that has as a diameter the line segment [0, n/m 2] on the positive x-axis, and a vertical line through the point where the circle and the parabola intersect above the x-axis.
Horner's method can be used to convert between different positional numeral systems – in which case x is the base of the number system, and the a i coefficients are the digits of the base-x representation of a given number – and can also be used if x is a matrix, in which case the gain in computational efficiency is even greater.
Graphs of y = b x for various bases b: base 10, base e, base 2, base 1 / 2 . Each curve passes through the point (0, 1) because any nonzero number raised to the power of 0 is 1. At x = 1, the value of y equals the base because any number raised to the power of 1 is the number itself.
If x kilograms of salami and y kilograms of sausage costs a total of €12 then, €6×x + €3×y = €12. Solving for y gives the point-slope form = +, as above. That is, if we first choose the amount of salami x, the amount of sausage can be computed as a function = = +.
The graph of the logarithm base 2 crosses the x-axis at x = 1 and passes through the points (2, 1), (4, 2), and (8, 3), depicting, e.g., log 2 (8) = 3 and 2 3 = 8. The graph gets arbitrarily close to the y-axis, but does not meet it. Addition, multiplication, and exponentiation are three of the most fundamental arithmetic operations.