i really dont get roots..ive tried all these problems but im not sure if im doing them rite?

Question

root 1/4 is 1/2?
root 1/16 is 1/4?
whats root 3/81? is it 1/9?
root a squared is a squared?
whats root A squared times A squared? its A to the fourth?
whats root A cubed?
whats root 2A to the third?

I mean square roots

root 1/4 is 1/2?
root 1/16 is 1/4?
whats root 3/81? is it 1/9?
root a squared is a squared?
whats root A squared times A squared? its A to the fourth?
whats root A cubed?
whats root 2A to the third?

Answer 1

It depends if you are talking about square roots or cube roots...

The first two look like square roots and are correct.

The third looks like a cube root but is incorrect. You would have to take the cube root of 3 (which is a decimal) and the cube root of 81, which is 3.

The root of a squared should be a, since taking a root and squaring are opposites and undo each other.

I believe a to the third is right, because of my previous comment.

Again, the last two depend if it's a cube root or a square root.

Answer 2

The way that you are using it, "root" means "square root". The term root can actually have a number of meanings in math.

When you take a square root (I'm going to abbreviate it sqrt from now on) you are looking for a number that when multiplied by itself results in the number that you were taking the square root of. For example sqrt(81) = 9 because 9 *9 = 81. And sqrt(121) = 11 because 11 * 11 = 121.

In your examples above, sqrt(1/4) = 1/2, which you can check by multiplying 1/2 * 1/2. Likewise, sqrt(1/16) = 1/4 because 1/4 * 1/4 = 1/16. If you have a fraction A/B then sqrt(A/B) = sqrt(A)/sqrt(B). So sqrt(3/81) = sqrt(3)/sqrt(81) = sqrt(3)/9

The square root of anything squared is simply the number.
sqrt(A^2) = A

Any time that you have multiple exponents you can simply multiply the exponents. For example (A^2)^3 = A^(2*3) = A^6. Since the square root can be expressed as a fractional exponent you have sqrt(A) = A^1/2. Notice that if we apply this property to the previous problem you get the same answer
sqrt(A^2) = (A^2)^1/2 = A^(2*1/2) = A^1 = A

The next problem illustrates a different property of exponents.
When you multiply two numbers that have the same base (A, in this case) you can simply add the exponents
sqrt(A^2)*A^2 = A*A^2 = A^1*A^2 = A^(1+2) = A^3


sqrt(A^3) simply goes back to the property of multiple exponents
sqrt(A^3) = (A^3)^1/2 = A^(3*1/2) = A^3/2

For the final one I think that you mean what is sqrt(2A^3). If that is the case then there is one more property of exponents that you need to know. The product of any two numbers raised to a power is the same as the product of the first number to that power times the second number to that power. More formally,
(A*B)^n = A^n * B^n

In this case we have sqrt(2A^3) = sqrt(2)*sqrt(A^3) = sqrt(2)*A^3/2

Please don't just copy my answers. Think about what I said. Email me [e-mail] if you have more questions (though I am very reluctant to simply do homework problems - been there, done that - if you don't understand what is going on I'm willing to explain more)