Arrange the following numbers in increasing order (smallest first, biggest last):
A = 2^(1/2)*4^(1/6)*8^(1/3)
B = 12*128
C = 8^(1/5)^2*8^(1/5)^3
D = 4*(-1)*2*(-1)*8*(-1)
E = 2^(1/2)*3*4^(1/4)
To arrange the numbers in increasing order, we need to simplify each expression first:
A = 2^(1/2) * 4^(1/6) * 8^(1/3)
= (√2) * (∛4) * (∛∛8)
= (√2) * (∛4) * (∛2)
= (√2) * 2 * (∛2)
= 2√2 * (∛2)
= 2√2 * ∛2
= 2√4
= 2 * 2
= 4
B = 12 * 128
= 1536
C = 8^(1/5)^2 * 8^(1/5)^3
= (8^(1/5)^2) * (8^(1/5)^3)
= (8^(2/5)) * (8^(3/5))
= 8^((2+3)/5)
= 8^(5/5)
= 8
D = 4 * (-1) * 2 * (-1) * 8 * (-1)
= (-4) * (-2) * (-8) * (-1)
= 64
E = 2^(1/2) * 3 * 4^(1/4)
= (√2) * 3 * (∜4)
= (√2) * 3 * (∜2)
= (√2) * 3 * 2^(1/8)
= (√2) * 3 * 2^(1/8)
≈ 1.68179
Now, let's arrange the numbers in increasing order:
4, 64, 8, 1.68179, 1536