Question

uniqueness Property

Answer 1

@precal do you by any chance know anything about uniqueness property

Answer 2

in simple english what they want you to do is write both sides with the same base
that is all
lets work through this one step by step

Answer 3

ok

Answer 4

\[625^{2-a}=5^{2}\] is the start right?

Answer 5

yep

Answer 6

the base on the right is \(5\) and the base on the left is \(625\)

Answer 7

now it turns out that \[5^4=625\]

Answer 8

so we can rewrite the left hand side of
\[625^{2-x}=5^2\] as
\[\large (5^4)^{2-x}=5^2\]

Answer 9

clear so far or no?

Answer 10

yes

Answer 11

k now we use the law of exponents on the left, that says when you raise to a power you multiply the exponents
rewrite \[\large (5^4)^{2-x}=5^2\] as
\[\large 5^{8-4x}=5^2\]

Answer 12

note the use of the distributive law here
since
\[4(2-x)=8-4x\]

Answer 13

so why does the 2 turn into a four istead of it being a -8 all together

Answer 14

oh

Answer 15

i jumped the gun but i hope i answered that question before you asked it
is it clear now?

Answer 16

yes

Answer 17

ok now come the famous "uniqueness property"

Answer 18

since
\[\large 5^{8-4x}=5^2\] that means that
\[8-4x=2\]

Answer 19

and you can solve that equation for \(x\) in two steps, which is only elementary algebra, has nothing to do with exponents

Answer 20

k

Answer 21

you want to try another one from that paper, or no?

Answer 22

yes please

Answer 23

\[36^x=216\] looks tricky but it is not hard

Answer 24

especially if you recognize
\[36=6^2\]and
\[216=6^3\]

Answer 25

oh actually they used a \(b\) no matter
\[36^b=216\\
6^{2b}=6^3\] then by "uniqueness"\[2b=3\]

Answer 26

k

Answer 27

that is all
good luck
nice picture btw
who drew it?

Answer 28

some person on google lol could you help me with one more please

Answer 29

I just need help with this last problem