Radical Within A Radical

[greatwhiteshark]

How do I solve a radical within a radical?

SAMPLE:

sqrt = square root

sqrtx^6 lies within a bigger radicand. Simplify.

I think the first steps is to multiply BOTH index. Is this true?

 

[Denis]

How do I solve a radical within a radical?

SAMPLE:

sqrt = square root

sqrtx^6 lies within a bigger radicand. Simplify.

I think the first steps is to multiply BOTH index. Is this true?

so you have: sqrt[sqrt(x^6)]
since x^3 * x^3 = x^6, then: sqrt(x^3)
since x^2 * x = x^3, then: sqrt(x^2 * x)
since sqrt(x^2) = x, then: x[sqrt(x)]

So answer is x times the square root of x : capish?

also, sqrt[sqrt(n)] = n^(1/4) : but I'm not getting into that...

 

[soroban]

Hello, greatwhiteshark!

How do I solve a radical within a radical?

SAMPLE: sqrt{x^6} lies within a bigger radicand. Simplify.

I think the first steps is to multiply BOTH index. Is this true?

If you're thinking what I think you're thinking . . . yes.

sqrt(x⁶) means (x⁶)^1/2 .= .x³
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . __
And the square root of <u>that</u> is: . (x³)^1/2 . = . x^3/2 . . . or: . √x³


We can do it one step: . [(x⁶)^1/2]^1/2 . = . x^{(6)(1/2)(1/2)} . = . x^3/2

 

[greatwhiteshark]

okay

I get now but I will need to learn additional radical with a radical questions to pass my course. More questions like this later this week.

 

[Denis]

In case this helps you...

sqrt[sqrt[sqrt[sqrt(65536)]]]

do most inner first: sqrt(65536) = 256
do next one; sqrt(256) = 16
do next one; sqrt(16) = 4
do final one; sqrt(4) = 2

Each "sqrt" really means "to the power (1/2)";
there's 4 of them in that example; so (1/2)(1/2)(1/2(1/2) = 1/16;
and 65536^(1/16) = 2

Did it help...or did it add to confusion?