Question

I'm still kind of having problems with exponents.
I really don't understand when there are fractions involved!
(1/4)^(2/3) or
8b^(1/3)
Please show work and thank you!

Answer 1

Also when there is negatives involved!
Here is an example from my work:
(125/64)^(-1/3)

Answer 2

Let's take the first one:
$$
\Large{
1^\frac{2}{3} = (1^{1/3})^2=(\sqrt[3]{1})^2\\
4^\frac{2}{3} = (4^{1/3})^2=(\sqrt[3]{4})^2\\
\left( \cfrac{1}{4}\right )^{2/3}=\cfrac{1^\frac{2}{3}}{4^\frac{2}{3}}=\cfrac{(\sqrt[3]{1})^2}{(\sqrt[3]{4})^2}=\left (\sqrt[3]{\cfrac{1}{4}}\right )^{2}
}
$$
Does this make sense?

Answer 3

Yeah(: Thanks, but my teacher doesn't like it when we leave is in that format. He likes when they're fractions (without exponents) and can you do the last example I just added?

Answer 4

A negative exponent is just 1/whatever. Like \(x^{-2}\) is just \(\frac{1}{x^2}\)

Answer 5

Yeah but I don't understand negative fractional exponents):

Answer 6

\[\large \left(\frac{125}{64}\right)^{-\frac{1}{3}}\]
the minus sign in the exponent means flip it
the 3 in the denominator means take the cubed root

Answer 7

so 5/4?

Answer 8

Do you mean taking the exponent of a fraction? Just flip the fraction over.

Answer 9

The negative exponent, I mean.

Answer 10

$$
\Large{
(8b)^{1/3}=\sqrt[3]{8b}
}
$$

Answer 11

close but you forgot to flip it

Answer 12

@ybarrap Could the answer be 4 times the cubed root of b?

Answer 13

$$
\Large{
(8b)^{1/3}=\sqrt[3]{8b}=2\sqrt[3]{b}
}
$$

Answer 14

work through the worksheet i sent, i bet if you do it step by step you will get it. look at the examples first

Answer 15

OH you flip the inside number not the exponent?

Answer 16

right!!

Answer 17

no one thinks for example that \(-\frac{2}{3}=\frac{3}{2}\)
if the exponent is negative, the notation means take the reciprocal (flip it)

Answer 18

Okay(: Um what if it's an integer and a fractional exponent? like: (25+144)^1/2

Answer 19

\[(\frac{a}{b})^{-1} = \frac{1}{\frac{a}{b}} = \frac{b}{a}\]

Answer 20

then don't flip it
leave it alone

Answer 21

Wait, it's the negative reciprocal of the exponent or inside number? @satellite73

Answer 22

just the reciprocal of the number inside
don't make it negative ever

Answer 23

Okay thanks(: @satellite73

Answer 24

\[\large 27^{-\frac{1}{3}}=\frac{1}{\sqrt[3]{27}}=\frac{1}{3}\] for example

Answer 25

Wow everyone is so nice(: Thanks! So when the inside number and exponents are fractions you just distribute the exponent and solve from then on?

Answer 26

\[\Large (\frac{ a }{ b })^{-\frac{ c }{ d }} = (\frac{ b }{ a })^{+\frac{ c }{ d }}\]

Answer 27

Okay, wow this is really hard. 32^(-3/5)?

Answer 28

$$
\Large{
\left (\cfrac{125}{64}\right )^{-1/3}\\
=\left (\cfrac{125^{-1}}{64^{-1}}\right )^{1/3}\\
=\left (\cfrac{64}{125}\right )^{1/3}\\
=\sqrt[3]{\cfrac{64}{125}}\\
=\cfrac{4}{?}
}
$$

Answer 29

? = 5 right? @ybarrap

Answer 30

yes

Answer 31

$$
\Large{
32^{-3/5}\\
(32^{-1})^{3/5}\\
\left (\cfrac{1}{32}\right )^{3/5}\\
=\cfrac{1}{(32)^{3/5}}\\
=\cfrac{1}{(\sqrt[5]{3})^3}
}
$$

Answer 32

There's no way to simplify that any more?

Answer 33

If \(\large(\sqrt[5]{3})^3=\sqrt[5]{3}\times\sqrt[5]{3}\times\sqrt[5]{3}\) can be simplified, then yes. But I don't see any way.

Answer 34

Okay! wow thanks everyone! (: I actually get it now (surprise)

Answer 35

yw