Question

One of your friends sends you an email asking you to explain how all of the following expressions have the same answer.
the cube root of x cubed

:x to the one–third power • :x to the one–third power • x to the one–third power

1 over x to the –1 power

the eleventh root of the quantity of x to the fifth times x to the fourth times x squared
Compose an email back assisting your friend and highlight the names of the properties of exponents when you use them.
Can u please help me out? @jim_thompson5910

Answer 1

\[\sqrt[3]{x^3}=x\] because, well because it is obvious
that is what the cubed root means

Answer 2

I know they all equal x I've already solved for all of them but Idk how to highlight the names of the properties of exponents.

Answer 3

idk what that's supposed to mean.

Answer 4

\[\large x^{\frac{1}{3}}\times x^{\frac{1}{3}}\times x^{\frac{1}{3}}=x^{\frac{1}{3}+\frac{1}{3}+\frac{1}{3}}=x^1=x\]

Answer 5

if you like exponential notation, then
\[\sqrt[3]{x^3}=x^{\frac{3}{3}}=x^1=x\]

Answer 6

\[\frac{1}{x^{-1}}\] the minus one in the exponent means take the reciprocal (flip it) and the reciprocal of \(\frac{1}{x}\) is \(x\)

Answer 7

did u just see what i wrote a couple minutes ago?

Answer 8

I know they all equal x I've already solved for all of them but Idk how to highlight the names of the properties of exponents.

Answer 9

ok for the one that says
\[\large x^{\frac{1}{3}}\times x^{\frac{1}{3}}\times x^{\frac{1}{3}}=x^{\frac{1}{3}+\frac{1}{3}+\frac{1}{3}}=x^1=x\] the property is that when you multiply you add the exponents

Answer 10

for the one that says
\[\frac{1}{x^{-1}}\] the property is that
\[b^{-1}=\frac{1}{b}\] i.e. the reciprocal, and the reciprocal of \(\frac{1}{x}\) is \(x\)

Answer 11

\[\sqrt[11]{x^5x^4x^2}=\sqrt[11]{x^{11}}=x\] the property is
1) when you multiply you add the exponents and
2) \(\sqrt[n]{x^n}=x\)

Answer 12

same property that explains why
\[\sqrt[3]{x^3}=x\]

Answer 13

tell your friend to stop emailing with such stupid questions

Answer 14

LOOOOOL!! ikr! who e-mails stuff like that these days anyways!

Answer 15

no one

Answer 16

hahaha seriously!

Answer 17

is it okay if you help me explain how I got all the same answer in complete sentences explaining properties as well plz?!

Answer 18

i thought i did above
do you see the explanation in each answer?

Answer 19

to recap
1) \(\sqrt[n]{x^n}=x\) for the the first one
in english, the nth root of x to the nth power is x

Answer 20

yeah but in complete sentences

Answer 21

2) when you multiply with the same base, you add the exponents
in the case one third plus one third plus one third is one and anything to the power of one is itself

Answer 22

its one third TIMES

Answer 23

3) the property is that raising a number or variable to the power of minus one means take the reciprocal
the reciprocal of one over x is x

Answer 24

not plus lol

Answer 25

yes, what i said
when you MULTIPLY you ADD the exponents

Answer 26

OH okayyy!!

Answer 27

and the last one \[\sqrt[11]{x^5x^4x^2}=\sqrt[11]{x^{11}}=x\] is a combination of two properties
when you MULTIPLY you ADD the exponents, and the nth root of x to the nth power is x

Answer 28

can u write however ur explaining right now for every single one of them plz?

Answer 29

i meant "i wrote them all in english, one at a time"

Answer 30

also labelled them 1), 2), 3), 4)

Answer 31

Thank you!!!!!!!!!

Answer 32

yw

Answer 33

ur awesome!