Question

2. Simplify the given expression to rational exponent form and justify each step by identifying the properties of rational exponents used. All work must be shown. 1/ (3 root x^-6)

Answer 1

@phi

Answer 2

@iambatman

Answer 3

@hartnn

Answer 4

is this
\[ \frac{1}{\sqrt[3]{x^{-6} }} \]

Answer 5

yes

Answer 6

you need to remember 3 things:
1) when you see "root" think EXPONENT IS A FRACTION
example: square root means exponent 1 /2
cube root means exponent is 1 /3
n th root means exponent is 1 / n

can you use that idea to rewrite
\[ \sqrt[3]{x^{-6}} \]
?

put parens around the "stuff" inside the radical sign, and put an exponent 1 /3
to the upper right of the parens

Answer 7

\[(\sqrt{-18x})^3\]

Answer 8

don't change what is inside the radical sign (the \( \sqrt{} \) sign)
in other words \(x^{-6}\) does not change into \( -18x\)

also, read carefully:
1) REMOVE the radical sign.
2) put parens around what's left
3) make the exponent 1 / 3 (notice 3rd root becomes exponent 1 /3

Answer 9

\[(x^{-6})^3\]

Answer 10

almost. but the exponent should be 1 /3 (that means the 3rd root)

Answer 11

so (x^-6)^1/3

Answer 12

yes. The important "rule" is change the radical to a FRACTIONAL exponent.

next we use rule #2
\[ \left(a^b\right)^c = a^{bc} \]
we MULTIPLY the exponents to get rid of the parens
can you do that ?

Answer 13

(x^-6)^1/3
x^-2
since -6 times 1/3 is 2

Answer 14

yes, but I would say
since -6 times 1/3 is -2

Answer 15

so far we have
\[ \frac{1}{\sqrt[3]{x^{-6} }} = \frac{1}{x^{-2} }\]

rule #3:
if we "flip" a fraction, we change the sign of the exponent:
\[ \frac{1}{a^{-b}} = a^b \]
or
\[ \frac{1}{a^{b}} = a^{-b} \]

can you use that rule on
\[ \frac{1}{x^{-2} }\]?

Answer 16

\[\frac{ x^2 }{ 1 }\]

Answer 17

yes, and anything divided by 1 is itself
in other words x^2 / 1 = x^2

Answer 18

so the answer is x^2

Answer 19

yes.

Hopefully if you learn the 3 rules, you can do any of these problems.

Answer 20

ok thank you very much phi

Answer 21

yw