How do you simplify #sqrt[81x^12y^8z^10]#?

2 Answers
Aldrich
Sep 9, 2015

#sqrt(81x^12y^8z^10)= 9x^6y^4z^5#

Explanation:

You can use the following rule:

#sqrt(ab) = sqrt(a) xx sqrt(b)#

In order to see how to simplify #sqrt(81x^12y^8z^10)#, we can split it as follows:

#sqrt(81x^12y^8z^10)#
#=sqrt(81) xx sqrt(x^12) xx sqrt(y^8) xx sqrt(z^10)#

Remember that #sqrt(a) = a^(1/2)#, so:
#sqrt(a^n)#
#=a^(n^(1/2))#
#=a^(n/2)#

In other words, square rooting an expression with an exponent halves the exponent.

#sqrt(81) xx sqrt(x^12) xx sqrt(y^8) xx sqrt(z^10)#
#= 9 xx x^6 xx y^4 xx z^5#

#= 9x^6y^4z^5#

Note: usually teachers will be fine with you skipping straight from #sqrt(81x^12y^8z^10)# to #9x^6y^4z^5#. You don't need to show the long process all the time unless your teacher asks for it.

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Don't Memorise
Sep 9, 2015

# sqrt[81x^12y^8z^10] = color(green)(9*x^(6)*y^(4)*z^(5)#

Explanation:

There are two simple Exponents rules we need to know to answer this question

1) #color(blue)(sqrt(a*b*c) = sqrta*sqrtb*sqrtc#

2) #color(blue)(sqrt(a^m) = a^(m/2)#

Based on the first rule,

#sqrt[81x^12y^8z^10] = sqrt 81*sqrt(x^12)*sqrt(y^8)*sqrt(z^10)#

# = sqrt(9^2)*sqrt(x^12)*sqrt(y^8)*sqrt(z^10)#

Based on the second rule, we write the above expression as

# = 9^(2/2)*x^(12/2)*y^(8/2)*z^(10/2)#

# = color(green)(9*x^(6)*y^(4)*z^(5)#

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