Question

Algebra 1 Help?

Answer 1

\[(3^{\frac{ 2 }{ 3 }})^{\frac{ 1 }{ 6 }}\]

Answer 2

Rewrite the rational exponent as a radical expression.

Answer 3

@hba @phi

Answer 4

First, use the rule
\[ (x^a)^b = x^{ab} \]
in other words multiply the exponents. Can you do that ?

Answer 5

\[\frac{ 1 }{ 9 }\]

Answer 6

?

Answer 7

Yes so now simplify this \[3^\frac{ 1 }{ 9 }\]

Answer 8

if you get a fraction like 1/9 that means the ninth root
do you know how to write that ?

Answer 9

Would it be: \[\sqrt[9]{3}\]

Answer 10

yes

Answer 11

Awesome! Thank You!

Answer 12

btw, if we got
\[ 3^{\frac{2}{9}} \]
we coud "undo the multiply" and write
\[ (3^2)^{\frac{1}{9}} \]
notice that we would change 3^2 to 9, and then write the ninth root of 9

Answer 13

How could we do it is for instance a question says:
\[\sqrt[3]{2}^{7}\]

Answer 14

So its kind of like working the same problem backwards except it is asking: Rewrite the radical as a rational exponent. Instead of the opposite.

Answer 15

But how would I figure it out?

Answer 16

the "root" part is a fraction with 1 on top
so ½ means square root
⅓ means cube root
¼ meant fourth root

if you have a fraction that does not have 1 as the top, then re-write the fraction

example: ⅔ becomes 2*⅓ (separate out the top)

Answer 17

So in this case would I use 3?

Answer 18

if you have, for example
\[ 4^{\frac{2}{3}} \]
we want to split the fraction to be 2 * ⅓
\[ 4^{2 \cdot \frac{1}{3}} \]
now we "undo the multiply"
\[ (4^2)^{\frac{1}{3}} \]

now we can write that as the third root (also called the cube root) of 4^2
or cube root of 16

Answer 19

I get that part. My problem is changing the radical into a fraction. What steps must I take?

Answer 20

if you have
\[ \sqrt[3]{x} \]
the exponent will be a fraction with 1 as its "top"
the 3 becomes the bottom

Answer 21

So if you have \[\sqrt[3]{x}^{2}\]

Answer 22

The the exponent will be a fraction with 2 at its top and 3 at the bottom?

Answer 23

change the \( \sqrt{} \) to parens, and make the exponent ⅓
\[ \sqrt[3]{x^2} = (x^2)^\frac{1}{3} \]
then multiply exponents (if you want to simplify even more)

Answer 24

If i multiply the exponents, then I will get \[x ^{6}\]

Answer 25

correct?

Answer 26

At least in this case..

Answer 27

almost, but the exponents are 2 and ⅓

we are using the rule
\[ (x^a)^b = x^{ab} \]
\[ (x^2)^\frac{1}{3} = x^{\frac{2}{3}} \]

Answer 28

the idea is rewrite the root part as an exponent that is a fraction (with a top = 1)

then, once you have exponents, you can multiply them

Answer 29

Okay. One more questions. I'm sorry, i'm just trying to cover all of my bases.

Answer 30

If I have a question such as, I don't know, maybe:

Rewrite the rational exponent as a radical by extending the properties of integer exponents.

\[\frac{ 2^{\frac{ 7 }{ 8 }} }{ 2^{\frac{ 1 }{ 4 }} }\]

Answer 31

How would I do that? I apologize for the multiple questions. I am just trying to make sure I know all aspects of these types of questions.

Answer 32

you need to know these rules
\[ x^a \cdot x^b = x^{a+b} \\ \frac{x^a}{x^b} = x^{a-b} \\ (x^a)^b= x^{ab} \]

Answer 33

when a and b are integers, these rules are *almost* obvious.
example
x*x = \(x^1 \cdot x^1= x^2 \)
\[\frac{x^2}{x}= \frac{x\cdot x}{x} = x= x^{2-1}\]

Answer 34

for this problem
\[ \frac{ 2^{\frac{ 7 }{ 8 }} }{ 2^{\frac{ 1 }{ 4 }} } = 2^{\frac{ 7 }{ 8 }-\frac{ 1 }{ 4 }}\]

Answer 35

you get
\[ 2^\frac{5}{8} \]

Answer 36

which you write as
\[ 2^{5 \cdot \frac{1}{8}}= (2^5)^\frac{1}{8} \]
or
\[ \sqrt[8]{2^5} =\sqrt[8]{32}\]