- #1
Asawira Emaan said:Kindly help me with this.
Solve
i^(-21/2)
Note: i means iota.
MarkFL said:I think I would begin by writing:
\(\displaystyle z=i^{-\frac{21}{2}}=\left(\cis\left(\frac{\pi}{2}\right)\right)^{-\frac{21}{2}}=\cis\left(-\frac{21\pi}{4}\right)=\cis\left(\frac{3\pi}{4}\right)=\frac{-1+i}{\sqrt{2}}\)
Does that make sense?
Klaas van Aarsen said:Hi Asawira Emaan, welcome to MHB!
Are you aware that $i=e^{\frac\pi 2 i}=\cos(\pi/2)+i\sin(\pi/2)$?
It means that we can calculate it as:
$$(i)^{-\frac{21}{2}} = (e^{\frac\pi 2 i})^{-\frac{21}{2}}
= e^{-\frac{21\pi}{4}i} = \cos\left(-\frac{21\pi}{4}\right)+i\sin\left(-\frac{21\pi}{4}\right)
$$
Asawira Emaan said:but the answer of this question is -i (negative iota)
Asawira Emaan said:but the answer of this question is -i (negative iota)
A negative fractional exponent represents the reciprocal or inverse of the base raised to the positive power of the fraction. In other words, it is the same as the base raised to the positive power of the fraction, but with the result being inverted.
To solve complex numbers with negative fractional exponents, you can use the rules of exponents and the properties of complex numbers. First, rewrite the negative fractional exponent as a positive exponent using the reciprocal property. Then, apply the rules of exponents to simplify the expression. Finally, use the properties of complex numbers to express the answer in the form of a complex number.
Yes, it is possible to have a negative fractional exponent for a complex number. This is because complex numbers can have both a real and an imaginary part, and the exponent can be applied to both parts separately. Therefore, the negative fractional exponent can be applied to the real or imaginary part of the complex number.
Yes, a negative fractional exponent can result in a real number. This can happen when the base is a negative number and the exponent is a rational number with an odd denominator. In this case, the result will be a real number because the negative number will be raised to an odd power, resulting in a negative value that can be simplified to a positive real number.
Negative fractional exponents are commonly used in engineering, physics, and other scientific fields to represent quantities that change over time or space. They are also used in financial calculations, such as compound interest, and in computer programming languages to represent complex numbers and perform calculations with them.