Simplify √125m^7

Question

I know how to simplify √125. That equals 5√5. I am confused when the m^7 is added to √125m^7

Answer 1

So, 125 = 5(5)(5) = 5²(5), so √((5²)5) = 5√5

 

In like fashion, √m⁷ = √((m⁶)m) = m³√m

 

√125m⁷ = 5m³√5m

Answer 2

This is why you learn fractional exponents!

 

the answer is (5√5)m^7/2  which is 5√5(m³m^½) which is simplified to 5m³√(5m)

Answer 3

I'm going to explain the whole thing, Jackson, even though you already get part of it. I'm doing this because this method works for everything under the radical.

First, prime factor anything you can. In this case, your 125 = 5³

Now, take the root, and remember to identify the index. For a square root, it's 2 of course:

²√ (5³ * m⁷)

Since a square root means you can only remove things in pairs, you're really asking how many times the 2 in the root divides into each power. For the base of 5, how many 2s can you get out of 3? 1 with a remainder of 1, so one 5 comes out and one remains:

 

5^{¹ 2}√ (5¹ * m⁷)

The base of m does the exact same thing! Just ask how many 2s are in 7. That would be 3, with one leftover, so you can bring out 3 of the Ms and leave one behind:

 

5m³ ²√ (5¹ m¹)

...or more simply:

5m³ √ (5m)

This works with every mix of numbers and variables, and with every integer root as well!