I really need help I don't have a math teacher so I'm trying learn this on my on so please someone help
What is 16 5/4 in simplest form?
There more questions but I'm putting the file down in the comments.

- anonymous

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- anonymous

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- phi

simplest form is a bit vague. But as a rule, if you see a fraction like
\[ \frac{5}{4}\] where the top is bigger than the bottom, that is an "improper fraction"
which means people would rather see it written as a mixed number
can you write 5/4 as a mixed number?

- anonymous

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- anonymous

i guess

- phi

5/4 means divide 4 into 5, plus a remainder that you "put over" 4

- anonymous

if you divide 4 into 5 it's 0.8

- phi

that is 4/5 = 0.8
in other words you did 5 divided into 4

- anonymous

either I'm stupid or i just don't get what you're saying

- phi

Can you post a copy of the question?

- anonymous

yes hold on

- anonymous

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- anonymous

What he's trying to say is you need to do the problem 5 divided by 4 as a starter

- phi

oh, that is different from what I thought
5/4 is an exponent

- anonymous

Oh same here :/

- anonymous

5 divided by 4 is 1.25

- anonymous

Ok so basically this problem is asking what is 16 to the power of 1.25 right @phi

- phi

yes, 5/4 = 1.25 but that is not the problem we have to solve here
the way we solve this is write the problem as
\[ \left( 16^\frac{1}{4}\right)^5 \]
the 1/4 power means the "fourth root"

- anonymous

Right

- phi

to find the 4th root (which is generally hard to do, but possible here)
we should factor 16 into
2*2*2*2 = \(2^4\)

- phi

do you see that 2 times itself 4 times is 16 ?

- anonymous

81?

- phi

so we write 16 as 2^4
\[ \left( 16^\frac{1}{4}\right)^5 = \left( (2^4)^\frac{1}{4}\right)^5\]

- phi

now use the rules of exponents
we use this rule
\[ (a^b)^c = a^{bc} \]
on
\[ (2^4)^\frac{1}{4} \]
can you do that ?

- anonymous

I'm stupid and lost

- anonymous

do 2x2x2x2

- anonymous

right?

- phi

the rule \( (a^b)^c = a^{bc} \)
means if you have an exponent b and another exponent c , we can multiply them

- phi

you "match the pattern"
\[ (a^b)^c = a^{bc} \\ (2^4)^\frac{1}{4} \]

- phi

we could do 2*2*2*2
but we don't want to (because if we use the exponent rule we will get a simpler answer)

- phi

look at these two things
\[ (a^b)^c = a^{bc} \\ (2^4)^\frac{1}{4} \]
do you see you can match a with 2, and 4 with b, and 1/4 with c ?

- phi

the idea is we can rewrite (2^4)^(1/4) using that rule
\[ 2^{4 \cdot \frac{1}{4} }\]

- anonymous

i see numbers that's makes no sense to why the hell letters are in a math problem and we are talking about 16 and a 5/4 so where in the world did a 2 and a 1 come form

- anonymous

I'm sorry that im getting mad and rude but i've been doing this all day and been have to learn this on my on

- phi

the letters are how to show a "rule"
we could use words, but it gets confusing.
anyway, we started with
\[ 16^\frac{5}{4} \]
we use a "rule" to write that a different way
\[ (16^\frac{1}{4})^5 \]
before going on, do you know what 1/4 * 5 is ?

- anonymous

1/4

- phi

\[ \frac{1}{4} \cdot 5 = ?\]
(as an improper fraction)

- anonymous

sorry 1.25

- phi

ok, but not as a decimal. what about as a fraction ?

- anonymous

no i dont i just know what it is as a decimal because my phone told me

- phi

when you multiply fractions, you multiply top times top and bottom times bottom
(if a number (like the 5) has no "bottom" , assume it is 1)
now try again
\[ \frac{1}{4} \cdot 5=?\]

- anonymous

5/4

- phi

yes. the reason we want to know that is
we can say
\[ \frac{5}{4}= \frac{1}{4} \cdot 5 \]
and vice versa

- phi

and there is a rule that let's us write
\[ 16^\frac{5}{4} = (16^\frac{1}{4})^5 \]

- anonymous

so now i know where you got 1/4

- phi

if you see
\[ (16^\frac{1}{4})^5\]
you should remember you are allowed to write it as
\[ 16^\frac{5}{4} \]
we need to be able to between these two different ways

- phi

so far we have
\[ (16^\frac{1}{4})^5\]
the next thing is to know we can write 16 as 2*2*2*2
(this is the hard part, knowing that. but now you do. (don't forget) )

- phi

do you know how to use the "short-cut" way using exponents to write
2*2*2*2 ?
in other words that is 2^?

- phi

2*2*2*2 is 2 to some power (some exponent)
do you know what little number we should put in the upper right of 2 so that it means 2*2*2*2 ?

- anonymous

the answer 17 1/4

- phi

do you know \(2^1 = 2 \)
and
\( 2^2 = 2\cdot 2\)
and
\(2^3 = 2\cdot 2\cdot 2\)
?

- anonymous

yes

- phi

ok, so how do we write
\[ 2\cdot 2\cdot 2\cdot 2= 2^? \]

- anonymous

2 to the 4 power

- phi

ok, so we know 16= 2*2*2*2 and that is 2^4

- phi

\[ (16^\frac{1}{4})^5 \\ ((2^4)^\frac{1}{4})^5
\]

- phi

now let's use the rule on
\[ (2^4)^\frac{1}{4} \]
remember we can multiply the exponents ?

- anonymous

so I multiply 2^4 by 1/4?

- phi

you multiply the exponents , so just 4*1/4 and that is the new exponent

- anonymous

1

- phi

that means \[ (2^4)^\frac{1}{4} = 2^1 \]

- phi

so now we have this
\[ (16^\frac{1}{4})^5 \\ ((2^4)^\frac{1}{4})^5 \\ (2^1)^5 \]
notice we can use the rule again to multiply the exponents on the last line

- anonymous

(2^1)5 is 10

- phi

no, not 10
it is (2^1)^5
i.e. \( (2^1)^5\)
multiply the exponents (that means 1 and 5)

- anonymous

5

- phi

yes, and that means 5 is the new exponent.
so (2^1)^5 = 2^5
ok ?

- anonymous

okay i get that

- phi

they may want you to multiply that out for the final answer
what is 2 times itself 5 times ?

- anonymous

32

- phi

yes, that is the answer

- anonymous

thank you so much for helping me and last question are you a math teacher?

- phi

no

- anonymous

you need to be

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