Question

How can I possibly visually interpret this?

Answer 1

no

Answer 2

So I know that \[2^3 = 2 * 2 * 2\]or 2 multiplied by itself 3 times
but what is \[\large 2^{3.4}\] I can't grasp the concept of raising something to a decimal.

Answer 3

is it 2*2*2*2.4 idk

Answer 4

^Nice try

Answer 5

what about 2+2 vs 2*2.4?

Answer 6

Well, I would interpret that is 2.4 + 2.4 = 4.8

Answer 7

what about 2.4*2.4?

Answer 8

or 2.4*2.4*2.4 ???

Answer 9

^That is \[2.4^3\]

Answer 10

oh

Answer 11

Have you tried doing it with fractional exponents instead of decimals?

Answer 12

Fractions? Haven't tried that for this case.

Answer 13

\[\Large 2^{3.4} = 2^{3+ \frac{ 2 }{5 }}\]

Answer 14

how about
\[2^{3.4}=2^{3.4/1}=2^{34/10}\]

Answer 15

you can actually use the laws of exponents.
2^2.4 = 2^2*2^0.4...
or as agent smith said, you get \(\sqrt[5]{2^{12}}\)

Answer 16

\[\Large 2^{3.4} = 2^{3}*2^{ \frac{ 2 }{5 }} = 2^3*\sqrt[5]{2^2}\]

Answer 17

tbh this is all way easier to visualize if you understand logarithsm...

Answer 18

the fact is that there is a specific meaning to exponential, which is not "repeated multiplication"
at this point it might not make sense, but the definition of \(b^x\) is \[\huge b^x=e^{x\ln(b)}\]

Answer 19

@inkyvoyd I do know logarithms...

Answer 20

there's a difference between knowing how to manipulate logarithmic expressions and actually being able to visualize and understand what logarithms are doing. Have you ever played with a slide rule?

Answer 21

with this you can make sense out of say \[\huge 5^{\sqrt2}=e^{\sqrt{2}\ln(5)}\]

Answer 22

@agent0smith Ah, I see how you got that.

Answer 23

Okee, I think reading over this, cleared up some of my confusion. Thanks to all those who replied!

Answer 24

hey @agent0smith want to be friends