simplify radicals (m1/4 n1/4)^2(m^2n^3)1/2

Question

Can you please show me how to do this step by step.

Answer 1

Hi, Denisha!

 

Let me rewrite the question to make sure that I understand it.  You are trying to simply the following expression:

 

(m^1/4n^1/4)² * (m²n³)^1/2

 

There are a couple things from middle school math that will help with simplifying this expression.  First, recall that when the same variables/numbers are being multiplied together, the exponents are added.  For instance, the product of x³x² is x⁵ (that is, x²⁺³).  Secondly, when an exponent is raised to another exponent, the two exponents are multiplied together.  For example, (x³)² = x^3*2 = x⁶.

 

With those ideas in mind, let's take a look at the math problem you're working on.

(m^1/4n^1/4)² * (m²n³)^1/2

 

Step 1: Distribute the 2 and the 1/2 exponents through to the exponents inside their set of parenthesis.

(m^1/4*2 n^1/4*2) * (m^2*1/2 n^3*1/2) <--parenthesis are optional, they don't change anything.

m^1/2 n^1/2 * m¹ n^3/2

 

Step 2: Since everything is being multiplied together and multiplication is commutative (order doesn't matter), rewrite the expression by grouping the same letters together next to each other.

m^1/2 n^1/2 * m¹ n^3/2 = m^1/2m¹n^1/2n^3/2

 

Step 3: Add the exponents of the variables which are the same.

m^1/2m¹n^1/2n^3/2

m^(1/2+1) n^(1/2+3/2)

m^3/2 n^4/2

m^3/2 n²

 

Now I'm not sure how the directions suggest leaving your answer (if you need to convert the fractional exponents back to radical form or not), but if you have to convert the fractional exponents back into radical form, remember that the numerator of the fractional exponent stays with the number/variable that goes underneath the radical symbol (that is, it's part of the radicand).  The denominator of the fractional exponent tells you what root you are taking (ie - a denominator of 2 means you're taking a square root; a denominator of 3 means you're taking a cubed root...and so on).

 

So this expression could be rewritten in radical form like this:

n^2*√(m³)

 

I hope this helps!

 

Patty