IS THERE AN ALGEBRA PRO OUT THERE TONIGHT? SOLVE THE EQUATION PLZ?

27^(2x)=9^(x+1)

27 can be written as 3^3

9 can be written as 3^2

(3)^(3*2x) = 3^(2(x+1))

as bases are equal,equate the powers

6x = 2(x+1)

6x = 2x+2

4x = 2

x = 2/4

x = 1/2

It's probably wanting you to use logs, partial fractions or something but...

quite a lot of maths(and engineering for that matter!) is about spotting forms (but you'll probably not have been taught about them per se, because that can gets into *seriously* heavy number theory!)

...

ie looking at that I remember 27 is just a different form of 3*9 which is just a different form of 3*3*3 which is just a different form of 3^3

ie 27=3*9=3*3*3=3^3

and 9=3*3=3^2

any time you see 27 you can substitute in 3^3 as it's the exact same "value" just written in a different form/way

so I'd rewrite that equation as

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

and simplify

remembering (a^b)^c=a^(b*c)

3^((3*2)x)=3^(2*(x+1)

3^(6x)=3^(2*(x+1)

That last bit looks a bit unweildy so lets get move a"2" from the index to the base of the power on both sides...

works out as

(3^2)^(3x)=(3^2)^(x+1)

9^3x=9^(x+1)

now we have an equation of the form a^b=a^c

as a is the same on both sides b=c

as such

3x=x+1

2x=1

x=0.5

lets see if that value works

27^(2*0.5)=9^(0.5+1)

27^(1)=9^(1.5)

27=27

Yes that seems to work out

Others saw a way to simplify the problem through clever observations. But that method is of limited utility, for example suppose the problem is: 29^(2*x) = 7^(x +1). Try using their method on that.

This equation calls for logarithms and yields a general solution.

M^(2*x) = p^(x + 1)

(2*x)*ln(M) = (x + 1)*ln(P)

(2*x)*ln(M) = x*ln(P) + ln(P)

(2*x)*ln(M) - x*ln(P) = ln(P)

x*[2*ln(M) - ln(P)] = ln(P)

x = ln(P)/[2*ln(M) - ln(P)]..... general solution

if;

M = 27

P = 9

x = ln(9)/[2*ln(27) - ln*(9)]

x = 2.197225/[2*3.295837 - 2.197225]

x = 2.197225/[6.591774 - 2.197255]

x = 2.197225/4.394449

x = 0.5000000 .............................. Answer

Oh, and the other equation I proposed above:

M = 29

P = 7

x = ln(7)/[2*ln(29) - ln(7)]

x = 0.406365

Much easier using logarithms, you don't have to use clever observations.

The equation I derived is the general solution.