Simplify the expression

Question

Simplify the expression. Assume that all variables represented positive numbers.

(8x^-3y⁶)^-1/3(xy^1/2) 

Answer 1

Hello, Cathryn from California

 

First of all, let me apologize about the mess, but I don't know how to write an exponent using my ipad.  

Ok, let me remind you of the rules

1) When you multiply, and the have the same base add the exponents. Ex. (2a)(a)= 2a^2

2) When you raise a power to a power, you multiply the exponents. Ex. 

(2^3)^2= 2^6= 64

3) The final answer cannot have negative exponents. Ex. Y^-3= 1/(y^3) and  1/(2^-3)= 2^3=8,

 

I am going to break down the problem. 8 can also be written as 2^3= since (2)(2)(2)=8.  Here we go

 

1) (2^3)^(-1/3)= 2^-1.  Multiply (3/1)(-1/3)=-1

2) (x^-3)^(-1/3)= x       Multiply (-3/1)(-1/3)=1 

3) (y^6)^(-1/3)= y^-2   Multiply (6/1)(-1/3)= -2

 

So, the first have of the problem we end up with (2^-1)(x)(y^-2).   I am now to to multiply this expression by (x)(y^1/2).  Again, I am going to break to down.  Remember, since I am going to multiply if they have the same base add the exponents.

 

1) 2^-1= 1/2.  

2) (x)(x)= x^2.  Add 1+1

3) (y^-2)(y^1/2)= y^-3/2.... Add -2+ 1/2= -4/2+ 1/2= -3/2

 

(x^2)/[(2)(y^3/2)].  This should be our final answer.

 

I hope this helps more than it confuses :). Don't forget to rate my answer.  

 

D. Y. T.

Answer 2

Use the following properties of exponents:

1. x^-a = 1/x^a
2. (x^ay^b)c = x^acy^bc
3. x^a/x^b = x^a-b

 

(8x^-3y⁶)^-1/3(xy^1/2)

 

Apply rule 2:

 

= (8^-1/3xy^-2)(xy^1/2)

 

Apply rule 1, and noting that 8^1/3 = 2 (2*2*2=8):

 

= (x/2y²)(xy^1/2)

 

Multiply out, apply rule 3:

 

= x²y^1/2/2y²

 

= x²/2y^3/2