why is the answer to this question no real root?
(-81)^3/4
Because -81^3 is a negative number. If you take a 1/4 root of any negative number (which is the same thing as taking a square root twice), you get imaginary or complex (real + imaginary) numbers.
the third power of -81 is a negative number
you can not take a square root of a negative number and get a real number.
(1/4) power is square root of square root.
To determine why the expression (-81)^3/4 does not have a real root, we need to understand the concept of exponentiation and the properties of real numbers.
The expression (-81)^3/4 involves raising a negative number to a non-integer power. Let's break it down step-by-step:
1. Start with the base number: -81.
2. Raise -81 to the power of 3/4.
To simplify this further, we can rewrite -81 as (-1) * (81) since -81 is the product of -1 and 81.
Now, let's apply the properties of exponents:
((-1) * 81)^(3/4)
Now, we distribute the exponent to each factor:
(-1)^(3/4) * 81^(3/4)
Let's first focus on the exponent (-1)^(3/4). The exponent of 3/4 means we need to take the fourth root of -1 and then raise it to the power of 3.
Taking the fourth root, we find that (-1)^(1/4) equals the complex number i. This means that (-1)^(3/4) is equal to i^3, which is -i.
Now, let's return to our simplification:
(-i) * 81^(3/4)
Since 81 is a positive real number, raising it to any power (including 3/4) will always result in a positive real number.
So, the expression (-81)^3/4 simplifies to -i times a positive real number.
Since there is no way to express a complex number multiplied by a positive real number as a single real number, we conclude that there is no real root to the expression (-81)^3/4.