Algebra with trig math problems?

Question

1) if 3^x = 54, then 3^(x-2) = ?

2) The surface area of a cube is 6x^2 - 36x + 54, what would the volume be of the same cube?

Answer 1

The difference of exponents is a quotient.

3^(x-2) = 3^x/3^2

Since 3^x = 54 we have 54/9 = 6.

Surface area = 6(x^2 - 6x + 9) = 6(x-3)^2

Since Surface area = 6(Side)^2, a side must be (x-3)

So, the volume is (x-3)^3

Answer 2

There are many ways you can do this. You can either find x and substitute:

3^x = 54..... take logs on both sides

x log3 = log 54
x = log54 / log3 = 3.63

then 3^(x-2) = 3^(3.63-2) = 3^1.63 = approximately 6

Or you could do
3^(x-2) = 3^x /3^2 = 3^x/9 = 54 /9 (remember 3^x = 54)
= 6
2nd method is probably what is expected.

2) remember that the surface area of a cube with side 's' would be 6 * s^2 and the volume would be s^3

look at what is given 6(x^2 - 6x + 9)

so your s^2 = x^2 - 6x + 9 = (x-3)^2

s = (x-3)

then your s^3 = (x-3 )^3