Solve & Check this Exponential Equation | 4^(x+2) + 4^(x+5) = 65

[govindashit]

Explention is to good to understand every weak student.
It's very helf full for me.
Thank you.

[CTJ]

you could skip the cross multiplication step and recognize that since both have 65 in the numerator then in order for them to be equal the denominators must be equal so therefore 1 = 4^(x+2) and therefore x= -2

[Viesto]

Instead of dividing both sides by 4^(x+2) you can factor 4^(x+2) from the left side. [4^(x+2)]*(1+4^3)=65.

[GodotWorld]

Very neat. I'm really enjoying this premath series, it's really got my gears turning on math puzzles.

I had another solution, a couple of solutions actually :)

1) substitute y=x+2 (I like this method of pulling out common factors)
4^(x+2) + 4^(x+5) = 65 becomes
4^y + 4^(y+3) = 65

2) since [ a^(n+m) = a^n • a^m ]
4^y + 4^(y + 3) = 65 becomes
4^y + 4^y • 4^3 = 65

3) factor out 4^y
4^y + 4^y • 4^3 = 65 becomes
4^y (1 + 4^3) = 65

4) solve the brackets
4^y (1 + 4^3) = 65 becomes
4^y(1+64) = 65 becomes
4^y (65) = 65

5) divide both sides by 65
4^y(65) = 65 becomes
4^y = 1

*Note*: at this point you could say y must be zero, since 4^y = 1 and we know [ a^0 = 1 ]
so y = 0 therefore x+2=0 therefore x=-2 **Solved**

6) But let's keep going for proof and fun, Remove the substitution y=x+2 so
4^y = 1 becomes
4^(x+2) = 1

7) using [ a^(n+m) = a^n • a^m ] again
4^(x+2) = 1 becomes
4^x • 4^2 = 1

8) divide both sides by 4^2
4^x = 1/(4^2)

9) However [ 1/(a^n) = a^-n ] So
4^x = 1/(4^2) becomes
4^x = 4^-2

At this point it's pretty obvious x = -2 **Solved**
to prove it though

10) We know [ a^n = n log a ] so
4^x = 4^-2 can be
x log 4 = -2 log 4

11) Divide both sides by log 4 canceling the log 4 on both sides
x log 4 = -2 log 4 becomes
x = -2 **Solved**

[gila]

It is obvious that even + even = odd is impossible. So it should be odd + even = odd. The only possible odd value for 4^y is 1. The smaller one has to be 1. so x+2=0 or x= -2. Check x+5 and it looks good.

[math]

said 65 = 4^3 + 1 = 4^3 + 4^0, so x=2+x+5= 3, x=-2, thanks for sharing

[darkomarkovic]

16*4^x+1024*4^x=65, so we have that 4^x=1/16, x=-2.
And your way to solve this equation is very interesting !
Best regards sir 👍

[devondevon]

x=-2 (x = negative 2) Answer
4^x +2 + 4^x+5 =65
4^x(4^2)(1 + 1 (4^3)=65 {factor out 4x^+2 which can be written as 4^x(4^2) }
4^x(4^2)( 1+64)=65
4^x(4^2)(65)=65
4^x(4^2) =65/65
4^x (4^2) =1
4^x(16) =1
4^x =1/16
4^x =1/4^2
4^x = 4^-2
x=-2 Answer

[philipkudrna]

Please show the solution using logs as well!

[HappyFamilyOnline]

Great explanation sir👍
Thanks for sharing😀

[rangaswamyks]

Super sir
Ypu could have taken4^2 as common factor
Than x^x+2=0

[xyz]

The right side can be written as 4^(x+2) (1 + 4^3) = 4^(x+2) (65), the rest is quite easy.

[robertchai]

4²(4^x) + 4⁵(4^x) = 65
1040(4^x) = 65
4^x = 1/16
4^x = 4^(-2)
x = -2

[dhrubajyotidaityari]

4 powered X(4²+4⁵)=65
4 powered x (16.65)=65
4 powered x=4^(—2)
x=—2

[varadarajcuram]

One could solve intutively by substitution.

[fongalex]

Thanks for the video. It is simple and I solve it in 10s

[ystech.edu.]

4^(x+2) + 4^(x+5)=65
Take commen 4^x we get,
4^x(4^2 +4^5)=65
4^x(16+1024)=65
4^x=65/1040
Taking log both side
xLog4=Log(65/1040)

Therefore Answer = -2 Only

[bernardopontes]

I did that question about an hour ago!

[Sarif_boy_amit_]

It took me only 7 sec to solve by taking 4x+2 as common and cancelling 65 on each side

[ibrahimalazooz]

أحسنتم هذا يسمى عندنا بالمعادلات الاسية