Rational exponents (was: Math Discussion)

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SUMMARY

The discussion centers on the mathematical expression (-64)^(3/2) and the absence of a real solution for this problem. Participants clarify that the square root of a negative number results in an imaginary number, specifically using the imaginary unit "i" where i² = -1. The conversation emphasizes the distinction between real numbers and imaginary numbers, confirming that (-64)^(3/2) does not yield a real number due to the negative base under the square root operation.

PREREQUISITES
  • Understanding of rational exponents
  • Familiarity with imaginary numbers and the imaginary unit "i"
  • Basic knowledge of square roots and their properties
  • Concept of real versus complex numbers
NEXT STEPS
  • Study the properties of rational exponents in depth
  • Learn about complex numbers and their applications
  • Explore the concept of imaginary numbers and their significance in mathematics
  • Review the mathematical operations involving square roots of negative numbers
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Students studying algebra, mathematics educators, and anyone seeking to understand the concepts of imaginary numbers and rational exponents.

goolalklk
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Homework Statement


(-64)^(3/2)

Homework Equations


None.

The Attempt at a Solution


There is no answer that can be reached and it is supposed not be a real number. I was wondering why that is. How is it that there is no "real" answer to this problem?
 
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goolalklk said:

Homework Statement


(-64)^(3/2)

Homework Equations


None.

The Attempt at a Solution


There is no answer that can be reached and it is supposed not be a real number. I was wondering why that is. How is it that there is no "real" answer to this problem?
##(-64)^{3/2} = [(-64)^{1/2}]^3##
Does that answer your question?
 
Alternatively: (−64)3/2 = √(-64)3

You're right in that there is no real solution. The square root of a negative number is an imaginary number, in which case you must use "i" to express √(-1).

i2 = -1 and √(-1) = i.
 
Last edited:
When it says "there is no real solution to the problem" it means there is no real number. Do you understand the difference between "real numbers" and "imaginary numbers"?
 
goolalklk said:
There is no answer that can be reached and it is supposed not be a real number. I was wondering why that is. How is it that there is no "real" answer to this problem?

The negative sign makes it so that the stated number isn't real.
 
FeDeX_LaTeX said:
The negative sign makes it so that the stated number isn't real.
Not exactly.

Real numbers can be positive or negative. However, taking the square root of a negative number does not give a real number result. Instead, imaginary numbers were invented to overcome this problem. The imaginary unit i is defined such that i2 = -1.
 

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