Question

What is the solution of the equation?
(x-2)^3/4 = 8

Answer 1

is that

\[\frac{ (x - 2)^{3} }{ 4 } = 8\]

or

\[(x - 2)^{3/4} = 8\]

Answer 2

the second one

Answer 3

all right

so the left hand side is raised to the 3/4 power. is there a way to invert that? here is an example:

\[y^{5/7} = 10 \rightarrow (y^{5/7})^{7/5} = 10^{7/5} \rightarrow y = 10^{7/5}\]

Answer 4

i honestly have no clue what to do..

Answer 5

raise both sides to the 4/3 power. that will cancel the exponents on the left side

Answer 6

so it would look like this: \[(x-2) = 8_{4}^{3}\]

Answer 7

we are raising both sides to the 4/3 power so . . .

Answer 8

\[(x - 2) = 8^{4/3}\]

Answer 9

sorry i got the exponent flipped around... So now what would be the next step?

Answer 10

We can focus on analyzing 8^(4/3). another way of writing 8 to the 4/3 is saying

the 3rd root of 8^4

\[8^{4/3} = \sqrt[3]{8^{4}}\]

Answer 11

and without using a calculator .. we could also take note that 8 = 2^3 and rewrite the expression as:

\[\sqrt[3]{8^{4}} = \sqrt[3]{(2^{3})^{4}} = \sqrt[3]{2^{3} * 2^{3} * 2^{3} * 2^{3} }\]

Answer 12

taking the cubed root of each factor here:

\[\sqrt[3]{2^{3} * 2^{3} * 2^{3} * 2^{3}} = 2 * 2 * 2 * 2 = 16\]

Answer 13

so, when it is all said and done

\[8^{4/3} = 16\]

and our equation that was (x - 2) = 8^(4/3) is now simplified to

(x - 2) = 16

Answer 14

and then you just add 2 to both sides for the final answer?

Answer 15

yep

Answer 16

thank you i appreciate it. i'm gonna write this in my notes so i can remember all that lol

Answer 17

i mean if you could use a calculator it would be a lot easier, but i wasnt sure so i took the long route