Question

@phi [attachment]

Answer 1

how far can you get ?

Answer 2

\[9a ^{4}b ^{5} \] ??

Answer 3

is that right ?

Answer 4

can you show your steps?

Answer 5

\[\sqrt{3^{4}*a ^{8}b ^{10}} \]
\[^{2/2}\sqrt{3a ^{4/2}*a ^{8/2}b ^{10/2}}\]

Answer 6

ok, pretty close. But they want the 4th root
\[ \sqrt[4]{stuff} \]

Answer 7

oh..

Answer 8

and when you "divide the exponents" by the root, you *drop the radical* sign.

remember
\[ \sqrt[4]{3^4 a^8 b^{10}}= \left(3^4 a^8 b^{10}\right)^\frac{1}{4} \]

Answer 9

when I do these, I switch to the "exponent" form. simplify. then switch back to radical form

Answer 10

\[3^{1/4} * a ^{3}b ^{5/2}\]

Answer 11

go slower. Do you understand this step:
\[ \sqrt[4]{3^4 a^8 b^{10}}= \left(3^4 a^8 b^{10}\right)^\frac{1}{4} \\
= \left(3^4\right)^\frac{1}{4}\left(a^8\right)^\frac{1}{4}\left(b^{10}\right)^\frac{1}{4}
\]

Answer 12

and now use the rule
\[ \left(a^b\right)^c = a^{bc} \]

Answer 13

is is that the final answer ?

Answer 14

not yet. But we have to leave some work for you, right?

Answer 15

im doing the best i can D:

Answer 16

so far we have
\[ \left(3^4\right)^\frac{1}{4}\left(a^8\right)^\frac{1}{4}\left(b^{10}\right)^\frac{1}{4} \]
now use the rule
\[ \left(a^b\right)^c = a^{bc} \]
to simplify just this part
\[ \left(3^4\right)^\frac{1}{4} \]

Answer 17

3 ?

Answer 18

match 3 with a, 4 with b, and c with ¼
the rule says we get 3^(4*¼) = 3^1 = 3

Answer 19

now do the (a^8)^(¼)
same rule

Answer 20

\[a ^{2}\] ?

Answer 21

yes

Answer 22

now the last part (b^10)^(¼)

Answer 23

\[b ^{5/2}\] ?

Answer 24

yes, but write 5/2 as a "mixed number" rather than as an improper fraction

Answer 25

or, better, as a the sum of a whole number and a fraction

Answer 26

im confused.. i got like 3 answers..

Answer 27

so far we have
\[ \left(3^4\right)^\frac{1}{4}\left(a^8\right)^\frac{1}{4}\left(b^{10}\right)^\frac{1}{4} = 3 a^2 b^\frac{5}{2} \]

but we should write 5/2 as 2 + ½ (which is the same thing as 5/2 , right?)

Answer 28

\[ \left(3^4\right)^\frac{1}{4}\left(a^8\right)^\frac{1}{4}\left(b^{10}\right)^\frac{1}{4} = 3 a^2 b^2 b^\frac{1}{2} \]

Answer 29

is something puzzling you ?

Answer 30

which still equals to \[3a^2b ^{5/2}\]

Answer 31

yes, but they probably want to simplify as much as possible
in other words be write
\[ b^\frac{5}{2} = b^{2 + \frac{1}{2}}= b^2 b^\frac{1}{2} \]

Answer 32

and the final answer (using radicals) is
\[ 3 a^2 b^2 \sqrt{b} \]

Answer 33

thanks. what about this one ?

Answer 34

let's do it step by step. first change 256 to 2^ some number

then rewrite the radical as ( stuff) ^ (¼)

Answer 35

to 2 ?

Answer 36

hows that ?

Answer 37

divide ?

Answer 38

we should factor 256 into its prime factors. (which is only 2)
in other words, 256= 2* 128= 2*2*64... keep going

Answer 39

16 ?

Answer 40

2x2x39 ?

Answer 41

are you asking how to divide 2 into 64?
we (so far) have
256= 2* 128 (check with a calculator)
but 128 is 2* 64, so
256= 2* 2*64

now factor 64 into 2 and another number

Answer 42

so then 2x2x2x32
then 2x2x2x2x16 ?

Answer 43

yes, but 16 can be simplified to 2*8

Answer 44

so then itll be 2x2x2x2x2x8

Answer 45

and (cutting to the chase!) 8 = 2x2x2

Answer 46

okayy

Answer 47

and 256= 2x2x2x2x2x2x2x2
and now we see why people invented exponents. It 's easier to type
256= \(2^8\)

Answer 48

so then itll be
\[2^{8}a ^{6}b ^{12}c ^{2}\]

Answer 49

yes, but with either a radical sign around it (and a little 4 to show 4th root)
or
put parens around it , and write to the ¼ power

Answer 50

\[ \left( 2^8 a^6 b^{12} c^2\right)^\frac{1}{4} \]
then multiply the ¼ times each exponent inside

Answer 51

\[4a ^{1/4}b ^{1/4}c ^{1/2}\]

Answer 52

sorry 4a^3/2

Answer 53

ok on the 4, the a and the c. re-do the b

Answer 54

b^3 ?

Answer 55

yes. what do we have so far (all together)?

Answer 56

\[4a ^{3/2}b ^{3}c ^{1/2}\]

Answer 57

looking good.

now write a^(3/2) using whole number plus a fraction

Answer 58

in other words divide 2 into 3 to get 1 with a remainder of 1
we write 3/2 = 1 + ½

Answer 59

1.5 ?

Answer 60

yes, but we want to use fractions (not decimals) because we will be changing back to "radical form" and we need a fraction to do that.

anyways, a^(3/2) = a^(1+ ½) = a^1 a^(½) or a x a^(½)

Answer 61

now (all together ) what do we have ?

Answer 62

1(1/2) ?

Answer 63

are you still working on the 3/2 = 1 + ½ ?
as posted above, we used that to rewrite
\[ a^\frac{3}{2} = a^{1 + \frac{1}{2}} = a^1a^\frac{1}{2}= a a^\frac{1}{2} \]

Answer 64

now that we have rewritten the a part, what do we have for the whole answer?

Answer 65

a^(3/2) ?

Answer 66

we want to finish the problem
\[ 4a ^{3/2}b ^{3}c ^{½} \\ 4a\ a^{1/2}b ^{3}c ^{1/2}\]
or after changing the order
\[ 4ab ^{3}\ a^{1/2}c ^{1/2}\]

any idea what next ?

Answer 67

\[4aa ^{1/2}b^3c ^{1/2}\]

Answer 68

We could be done here, but they probably want you to replace the fraction exponents with radicals.

Answer 69

yeah ..

Answer 70

which is why I wrote the answer in this order
\[ 4ab ^{3}\ a^{½} c ^{½} \]
the a^½ turns into a square root. same for the c^1/2

Answer 71

which im still confused..

Answer 72

are you confused how to switch between radical and exponent form?

Answer 73

yes..

Answer 74

First, the two forms mean the same thing.
if you have a radical sign with a little number "n" out front
\[ \sqrt[n]{x} = x^\frac{1}{n} \]
you can write it as the exponent 1/n

If they don't show a little number n out front, it is assumed to be 2

Answer 75

to switch from x^(1/n) you write \( \sqrt{x} \) in its place, and then put a little "n" out front: \( \sqrt[n]{x}\)

Answer 76

\[ 4ab ^{3}\ a^{½} c ^{½} \]
becomes
\[ 4ab ^{3}\ \sqrt{a} \sqrt{ c} \]
and we can combine the square roots
\[ 4ab ^{3}\ \sqrt{ac} \]

Answer 77

so is that it ?