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scott_alexsk
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Hello,
How would you determine the value of such numbers as -3^1.5?
Thanks,
-scott
How would you determine the value of such numbers as -3^1.5?
Thanks,
-scott
As you wrote it, you do not have a square root of a negative number:Integral said:[tex] - 3 ^{ 1.5} = - 3^ {\frac 3 2} [/tex]
Do either the square root or the cube first, either way results in the square root of a negative number. So the solution lies in the imaginary plane.
[tex](-3)^{1.5} = (-3)^{\frac{3}{2}} = \sqrt{-27} = \sqrt{27} \sqrt{-1} = \boxed{3i \sqrt{3}}[/tex]BoTemp said:Have you ever worked with complex numbers before? Define i = (-1)^0.5
-3^1.5 = (3)^1.5 * (-1)^1.5 = 3^1.5 * (-1)^1 * (-1)^0.5 =
[tex] -i3\sqrt{3}[/tex]. By extension of this procedure one can define negative numbers to any power.
A good point!bomba923 said:As you wrote it, you do not have a square root of a negative number:
[tex]-3^{1.5} = -3^{\frac{3}{2}} = - \sqrt{27} = - 3\sqrt{3}[/tex]
On the other hand,
[tex](-3)^{1.5} = (-3)^{\frac{3}{2}} = \sqrt{-27} = 3i \sqrt{3}[/tex]
While "the" square root of a positive real number is define to be the positive root, that is not true for complex numbers, where most functions are "multi-valued". (-3)1.5 has two values,[tex](-3)^{1.5} = (-3)^{\frac{3}{2}} = \sqrt{-27} = \sqrt{27} \sqrt{-1} = \boxed{3i \sqrt{3}}[/tex]
not
[tex]-3i \sqrt{3}[/tex]
?? or even if k is rational: 1.5 is certainly rational! Did you mean*Also, note that:
[tex]\forall x < 0, x^k \notin \mathbb{R} \; \text{ if } \, k \notin \mathbb{Q}[/tex]
Not really ~~HallsofIvy said:You posted this 9 times?! I deleted the other 8.
I see; so the answer is thenHalls of Ivy said:While "the" square root of a positive real number is define to be the positive root, that is not true for complex numbers, where most functions are "multi-valued". (-3)1.5 has two values,
[itex]3i\sqrt{3}[/itex] and [itex]-3i\sqrt{3}[/itex]
I didn't intend to be an 'if and only if' statement...Halls of Ivy said:or even if k is rational: 1.5 is certainly rational!
Not really//Halls of Ivy said:Did you mean [tex]k \notin \mathbb {I}[/tex]?
~That's exactly what I did! (..."nine times" )HallsofIvy said:My reader is so messed up it doesn't always show the changes even after "refresh"! Sometimes what I do is make the changes, copy it, then delete the post (clicking "physically remove") and paste into a new post.
A negative number is a number less than zero, typically represented with a minus sign (-) in front of it. It is the opposite of a positive number.
A non-integer power is a number that is not a whole number or a fraction, but instead is a decimal or irrational number. It is used to indicate the number of times a number is multiplied by itself.
Yes, you can raise a negative number to a non-integer power. The result will depend on the specific numbers being used, but it is possible to get both positive and negative results.
The result will be either positive or negative, depending on the power being used. For example, if a negative number is raised to a power of 0.5, the result will be the square root of that number, which can be either positive or negative.
To calculate a negative number to a non-integer power, you can use a calculator or perform the calculation by hand. First, rewrite the number as a fractional power, then use the rules of exponents to simplify it. For example, (-2)^0.5 can be rewritten as (-2)^1/2, and then simplified to the square root of -2, which is an imaginary number.