Difficult indices questions!?

Question

Hi.
I've got some really hard homework on indices and equations, there are 3 questions that I'm particularly stuck on.. Even if you could answer one of them, it would be a great help - and please don't use logarithms:
i) 0.25 = 4^x-2
ii) 1/9 = 3^(x/2)
iii) 0.125 = (1/4)^-x

Thanks!

Answer 1

Question i)
1/4 = 4/x²
x² = 16
x = 4

Question (ii)
1/9 = 3^(x/2)
3^(- 2) = 3^(x/2)
x/2 = - 2
x = - 4

Question (iii)
1/8 = 4^x
2^(-3) = (2²)^x
2^(-3) = 2^(2x)
2x = - 3
x = - 3/2

Answer 2

Not using Logs makes it a bit harder, but by no means impossible, my arithmetic isn't what it used to be, but no matter. Just make sure you get comfortable with logs they're unbelievably handy. The alternative is making the bases the same.

I'm assuming you know all the rules of indicies to begin with, but I'll list them here for you:

a^0 = 1
(a^n)^m = a^nm
a^-n = 1/a^n
a^1/n = n(root)a
a^m/n = n(root)a^m
a^n x a^m = a^n+m
a^n/a^m = a^n-m

1) express 0.25 as a power of 4.

to me 0.25 = 1/4 which is 4^-1 by rules of indicies now,
x - 2 = - 1
x = 1

2) 1/9 = 3^(x/2)

Express 1/9 as 9^-1 = 3 ^(x/2)
now express 9^-1 as a power of 3, to make the bases the same, so now you have.

(3^2)^-1 = 3^(x/2)
3^-2 = 3^(x/2)
now that the bases are the same:
x/2 = - 2
x = 2(-2)
x=-4

3) 0.125 = (1/4)^-x
turn the power on the (1/4)^-x to a +x by taking the reciprocal, so you have 4^x.

4^x = 1/8, which is the same as 0.125

now 4^x = 8^-1

now ot make the bases the same it looks like a good idea to express both sides as powers of 2.

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

2x = - 3
x = -3/2
x = -1.5

Hope that helped, I tried to include all the working. Most people seem to know what they're talking about and they've got the right answer but the process by which you arrive at the answer in your head is also important, so I always suggest writing out the obvious.

Answer 3

The easiest way to do these is with logarithms.

But if you're not supposed to use 'em, you can change the equations around so that each side is an exponent (or power) of the same number. For example,

1/9 = 3^(x/2)
1/3^2 = 3^(x/2)
3^(-2) = 3^(x/2)
-2 = x/2
-4 = x.

Answer 4

1)

0.25 = 4^(x-2)

25/100 = 4^(x-2)

1/4 = 4^(x-2)

write 1/4 = 4^(-1)

4^-(1) = 4^(x -2)

x - 2 = -1 (powers are equal when bases are same)

x = 1

2)

1/9 = 3^(x/2)

3^(-2) = 3^(x/2) (since 1/9 = 1/3^2 = 3^(-2))

x/2 = -2

x = -4

3)

0.125 = (1/4)^(-x)

125/1000 = (1/4)^(-x)

1/8 = (1/4)^(-x)

1/2^3 = 1/2^(-2x)

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

2x = -3

x = -3/2

Answer 5

0.25 = 1/4 =4^-1= 4^-x^2 so x^2=1 and x =+-1
1/9=3^-2=3^x/2 so x/2=-2 and x= -4
0.125 =1/2^3 = 1/2^-2x so 3=-2x and x=-3/2
If the first exponent is x-2 you get x-2=-1 and x=1

Answer 6

sorry (i) cannot be solved without logarithms, unless you use a graphical soln which is not very accurate
unless the question reads
i) 0.25=4^(x-2)
0.25=4^-1
Hence 4^-1=4^(x-2)
=>-1=x-2
x=1
ii)1/9=3^-2
Hence3^-2=3^(x/2)
-2=x/2
x=-4
ii) 0.125=2^-3 and 1/4= 2^-2
hence
2^-3=(2^-2)^-x
2^-3=2^(2x)
Hence -3=2x
x=-3/2

Answer 7

i) .25 = 4^(x-2)

1/4 = 4^(x-2)
4^-1 = 4^(x-2)
-1 = x-2
x = 1


ii) 1/9 = 3^(x/2)
9^(-1) = 3^(x/2)
3^(-2) = 3^(x/2)

-2 = x/2
x = -4


iii) .125 = (1/4)^-x
1/8 = 1/4^-x
8^-1 = 4^(-)(-x)
2^-3 = 4^x
2^-3 = 2^2x
-3 = 2x

x = -1.5

Answer 8

0.25=4^(x-2)
1/4=4^(x-2)
4^-1=4^(x-2)
-1=x-2
-1+2=x
1=x

1/9=3^(x/2)
1/3^2=3^(x/2)
3^-2=3^(x/2)
-2=x/2
-4=x

0.125=(1/4)^-x
1/8=(1/4)^-x
1/2^3=(1/2^2)^-x
1/2^3=1/2^-2x
3= -2x
3/-2=x
-3/2=x

Answer 9

i) x = 0.584963
ii) x = -4
iii) x = -1.5