find the three geometric means between -2/3 and -54

hint: r = 3

meko nahi ata b.c

To find the three geometric means between two numbers, we need to find a common ratio between successive terms and then apply the formula for geometric mean.

First, let's find the common ratio between -2/3 and -54. The common ratio, denoted by "r", is found by dividing the second term by the first term:

r = (-54) / (-2/3)

To divide fractions, we multiply the first fraction by the reciprocal of the second fraction:

r = (-54) * (3 / -2)
r = 162 / -2
r = -81

Now, we can use the formula for geometric mean (denoted by "G") to find the three geometric means:

G1 = √((-2/3) * (-81))
G2 = √((-2/3) * (-81) * (-81))
G3 = √((-2/3) * (-81) * (-81) * (-81))

Let's calculate these values:

G1 = √((-2/3) * (-81))
= √(162/3)
= √54
= 2√3

G2 = √((-2/3) * (-81) * (-81))
= √(6561/3)
= √2187
= 3√3

G3 = √((-2/3) * (-81) * (-81) * (-81))
= √(531441/3)
= √177147
= 6√3

Therefore, the three geometric means between -2/3 and -54 are 2√3, 3√3, and 6√3.