Question

Part A: Use the properties of exponents to explain why 16 raised to the power of 1 over 4 is called the fourth root of 16. (5 points)

Part B: The length of a rectangle is 5 units and its width is square root of 5 unit. Is the area of the rectangle rational or irrational? Justify your answer. (5 points)

Answer 1

@campbell_st

Answer 2

well in index from

16 = 2^4

and is that is raised to the 1/4 power

\[(2^4)^\frac{1}{4} = 2^?\]

what happens..?

use the index law for power of a power

Answer 3

I'm Sorry But What?

Answer 4

@aaronq

Answer 5

well start by rewriting 16... with a base of 2, what is the power...?

Answer 6

2.....

Answer 7

no 2 is the base... so

\[2^? = 16\]

what value does ? have

Answer 8

1

Answer 9

or is it 8

Answer 10

no

\[2^1 = 2~~~and~~~~2^8 = 256\]

you are using powers

so what power might you use...

Answer 11

i do not know .

Answer 12

4

Answer 13

great so

\[2^4 = 16\]

so raising 16 to the power of 1/4 means

\[(16)^\frac{1}{4}\]

is you rewrite 16... using 2^4 the problem becomes

\[(2^4)^{\frac{1}{4}}\]

now when working with powers

fraction powers mean radicals

so

\[16^{\frac{1}{4}} = \sqrt[4]{16}\]

and there is an index law that says when finding the power of a power... multiply the powers

so

rather than use radicals

\[(16)^{\frac{1}{4}} = (2^4)^{\frac{1}{4}} = 2^{4 \times \frac{1}{4}}\]

so if you multiply the powers what do you get...?

Answer 14

the problem up top? right

Answer 15

no just multiply the powers in

\[2^{4 \times \frac{1}{4}}\]

Answer 16

0.125

Answer 17

4 X 1/4 = 1

So it simplifies to 2^1 =2

and when you pick up a calculator

\[\sqrt[4]{16} = 2\]

Answer 18

for the 2nd question

Area = length x width

so is

\[A = 5 \times \sqrt{5} = 5 \sqrt {5}\] a ration number of an irrational number

Answer 19

irrational

Answer 20

and you have the justification in the calculation

Answer 21

yes

Answer 22

i got 2 more question :( @campbell_st

Answer 23

post them as new questions... and someone will help

Answer 24

ok