Question

Question #2bca3

Answer 1

See below

Explanation:

Laws:
1) `x^(1/2)=sqrt(x) iff sqrt(x)=x^(1/2)`
2) `x^a*x^b=x^(a+b) iff x^(a+b)=x^a*x^b`
2) `x^a/x^b=x^(a-b)`

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`2/sqrt(2)=?`

First:
`2=2^1`
`=> 2^1=2^((1/2+1/2))`
`=> 2^(1/2+1/2)=2^(1/2)*2^(1/2)`

Second:
`2^(1/2) iff sqrt(2)`

and therefore:
`2/sqrt(2)=(2^(1/2)*2^(1/2))/(2^(1/2))=2^(1/2)=sqrt(2)`

Answer 2

See below.

Explanation:

`2/sqrt2` has a radical in its denominator. If we want to remove the radical, we can multiply the expression by `sqrt2/sqrt2`. This is essentially the same as multiplying by `1`, so we aren't changing the value of the expression.

`2/sqrt2 *sqrt2/sqrt2`

`sqrt2 * sqrt2` is simply `2`. Thus, we have

`(2sqrt2) / 2`

We can now cancel the `2` in the numerator and denominator.

`(cancel2sqrt2) / cancel2 = sqrt2`