How do you simplify sqrt(144x^6)√144x6?

2 Answers
Daniel L.
Oct 10, 2015

This can be written as 12x^312x3

Explanation:

sqrt(144x^6)=sqrt(144)*sqrt(x^6)=sqrt(12*12)*sqrt(x^6)=12*sqrt(x^6)=12x^3√144x6=√144⋅√x6=√12⋅12⋅√x6=12⋅√x6=12x3

The final step: sqrt(x^6)=x^3√x6=x3 comes from the properties of exponents, which say, that:

  1. root(n)(a)=a^(1/n)n√a=a1n
  2. (a^b)^c=a^(b*c)(ab)c=ab⋅c

Using the properties above we can write, that:

sqrt(x^6)=(x^6)^(1/2)=x^(6/2)=x^3√x6=(x6)12=x62=x3

George C.
Oct 10, 2015

sqrt(144x^6) = 12 abs(x^3)√144x6=12∣∣x3∣∣

Explanation:

If a, b >= 0a,b≥0 then sqrt(ab) = sqrt(a)sqrt(b)√ab=√a√b

If a >= 0a≥0 then sqrt(a^2) = a√a2=a

sqrt(144x^6) = sqrt(144)sqrt(x^6) = sqrt(12^2)sqrt(abs(x^3)^2) = 12 abs(x^3)√144x6=√144√x6=√122√∣∣x3∣∣2=12∣∣x3∣∣

The modulus operation is necessary to deal with the case x < 0x<0.

Going deeper:

Note that if a < 0a<0 then sqrt(a^2) = -a√a2=−a, so in general we can write sqrt(x^2) = abs(x)√x2=|x|.

Any number has two square roots. If a >= 0a≥0 then it has a positive square root: sqrt(a)√a and a negative square root: -sqrt(a)−√a.

If a < 0a<0 then it has square roots i sqrt(-a)i√−a and -i sqrt(-a)−i√−a. We call i sqrt(-a)i√−a the principal square root, defining sqrt(a) = i sqrt(-a)√a=i√−a.

When it comes to square roots of negative numbers, the identity sqrt(ab) = sqrt(a)sqrt(b)√ab=√a√b breaks down. For example:

1 = sqrt(1) = sqrt(-1 xx -1) != sqrt(-1) xx sqrt(-1) = -11=√1=√−1×−1≠√−1×√−1=−1

Related questions
See all questions in Simplification of Radical Expressions
Impact of this question
2994 views around the world
You can reuse this answer
Creative Commons License