Square root of x to the power of 9 and y to the power of 8

Question

Sqaure root

Answer 1

For the first one.

 

√(x⁹)

 

When we are dealing with radicals such as square-roots, cube-roots, 4th-roots, etc, we can break them up by their factors to simplify them.

 

√(x²) * √(x²) * √(x²) * √(x²) * √(x)

 

 

If we take the square of a number and then take the square-root of the result, we get the base number.  Here is why.

 

√(6²) = √(36) = 6

 

 

Therefore,

 

√(x⁹) = x⁴ * √x

 

 

 

For the second one.

 

y⁸

 

This one cannot be simplified any further.

Answer 2

Square root of (x to the power of 9) and square root of (y to power of 8).

 

Let's do the last first.  SQRT(y⁸) = y⁴       like SQRT(x²) is x, just divide the exponent by two for square root.

 

For SQRT(x⁹), we usually factor squares first making this SQRT(x⁸ * x)

                                  which results in x⁴ * SQRT(x)

Answer 3

SQRT is the same as putting something to the half power.

So SQRT x^9 = (x^9)^(1/2)

When you have a power to a power, you multiply the powers

....= x^(9/2)

Which can be simplified to = x(4 + 1/2)

We know that x^a  *  x^b  = x^(a+b), so the reverse is also true.

x^(a+b)= x^a * x^b 

x(4 + 1/2) = x^4  *   x^(1/2)

= (x^4)(SQRT x)