Identify all real sixth roots of 262,144 2.) Identify all real square roots of -64
The Correct Answer and Explanation is:
1. Sixth Roots of 262,144:
To find the sixth roots of 262,144, we first express 262,144 as a power of 2:262,144=218262,144 = 2^{18}262,144=218
The sixth roots are the values of xxx that satisfy:x6=262,144=218x^6 = 262,144 = 2^{18}x6=262,144=218
This implies that x=218/6=23=8x = 2^{18/6} = 2^3 = 8x=218/6=23=8. Thus, the principal sixth root is 888. However, there are six possible real roots for a sixth root because the equation x6=218x^6 = 2^{18}x6=218 can have multiple solutions due to the periodic nature of complex roots. Specifically, the sixth roots of a positive number rrr are given by:xk=r1/6⋅ei(2kπ/6)x_k = r^{1/6} \cdot e^{i(2k\pi/6)}xk=r1/6⋅ei(2kπ/6)
For real roots, the possible values are 888 (since we’re dealing with a positive number and the principal root is real) and the negative of the principal root, −8-8−8.
Thus, the real sixth roots of 262,144 are:8 and −8.8 \text{ and } -8.8 and −8.
2. Square Roots of -64:
For square roots of a negative number, we use the imaginary unit iii, where i=−1i = \sqrt{-1}i=−1.
We can write −64-64−64 as:−64=64×(−1)-64 = 64 \times (-1)−64=64×(−1)
The square roots of 646464 are 888 and −8-8−8. So, the square roots of −64-64−64 are:±8i\pm 8i±8i
Thus, the real square roots of −64-64−64 do not exist, but the complex square roots are:8i and −8i.8i \text{ and } -8i.8i and −8i.
Summary:
- The sixth roots of 262,144 are 888 and −8-8−8.
- The square roots of −64-64−64 are 8i8i8i and −8i-8i−8i, which are purely imaginary.
