Question

Vanessa and William are stuck simplifying radical expressions

Answer 1

That is Vanessa's^

Answer 2

\[16√x*x ^{3}*x ^{4}\]
Williams^

Answer 3

Using full sentences, describe how to fully simplify Vanessa and William's expressions. describe if Vanessa and William started with equivalent expressions or if they started with expressions that are not equal.

Answer 4

I'd start by getting the first thing into a single expression with a common base using \(\large \frac{x^a}{x^b}=x^{a-b}\)

Answer 5

I subtracted them and got 1/2.. is that right?

Answer 6

looks good, yeah, you can simplify x^1/2 further

Answer 7

You can?

Answer 8

fractional exponents can be written as roots

Answer 9

for example \(x^\frac{1}{3}=\sqrt[3]x\)

Answer 10

So this would be \[\sqrt[2]{x}\]

Answer 11

which is the same thing as \(\sqrtx\)

Answer 12

\(\sqrt x\)

Answer 13

Alright, so what about William's? How would that one be solved?

Answer 14

rewrite the square root
use the fact t hat \(x^a*x^b=x^{a+b}\)

Answer 15

\[x ^{3+4} = x ^{7}\]
right?

Answer 16

so far, but then there's the \(\sqrt x=x^\frac{1}{2}\)

Answer 17

Since there is a sixteen wouldn't it be \[\sqrt[16]{x} =x ^{7+16} = x ^{23}\]

Answer 18

Or is that wrong? Cause I'm kinda lost :/

Answer 19

is that \(16\sqrt x\) or \(\sqrt[16]x\)

Answer 20

there's a difference. the first one is 16*the square root of x
the second one is \(x^\frac{1}{16}\) or the 16th root of x

Answer 21

as I see it, \(\large 16\sqrt x*x ^{3}*x ^{4}=16x^\frac{1}{2}*x ^{3}*x ^{4}=16x^{4+3+\frac{1}{2}}=16x^\frac{15}{2}\)

Answer 22

It's \[\sqrt[16]{x}\] and I'm not sure how to solve it

Answer 23

again, roots can be written as fractional exponents and vice versa
\(\huge \sqrt[\color{green}{16}]x=x^\frac{1}{\color{green}{16}}\)

Answer 24

so in our case it'd be \(x^\frac{{\color{red}{1}}}{\color{red}{16}}*x^{4+3}=x^{7\frac{1}{16}}\)

Answer 25

And that is the complete simplified form? Sorry they didn't tell me how to solve that one

Answer 26

yeah, so vanessa: \(x^\frac{1}{2}=\sqrt x\)
william's thing: \(x^{7\frac{1}{16}}\)

Answer 27

Thank you so much :)

Answer 28

any time :)