How do you simplify # sqrt ((4a^3 )/( 27b^3))#?

2 Answers
Jun 9, 2016

#sqrt((4a^3)/(27b^3))=(2a)/(3b)sqrt(a/(3b))#

Explanation:

#sqrt((4a^3)/(27b^3))#

= #sqrt((2xx2xxaxxaxxa)/(3xx3xx3xxbxxbxxb))#

= #sqrt((ul(2xx2)xxul(axxa)xxa)/(ul(3xx3)xx3xxul(bxxb)xxb))#

= #(2a)/(3b)sqrt(a/(3xxb))#

= #(2a)/(3b)sqrt(a/(3b))#

Jun 10, 2016

Just another way of writing the same thing as the other solution:

#" "color(green)((2asqrt(3ab))/(3b^2))#

Explanation:

Given:#" "sqrt((4a^3)/(27b^3))#

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Concept of approach")#

You look for numbers that are squared. Take them outside the square root and 'get rid' of the square.

Example: Suppose we had #sqrt(12)#

Choosing the factors of #3xx4=12# we write it as:

#sqrt(3xx4)# but 4 is #2^2# giving #sqrt(3xx2^2)#

Take the 2 out side the square root giving

#" "2sqrt(3)#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Solving your question")#

#color(brown)("Mathematicians do not like square roots in the denominator so we will 'get rid' of it")##color(brown)("later.")#

#" Write as: "(sqrt(4a^3))/(sqrt(27b^3))#
'.......................................................................................
we know that
#27" "=" "3xx9" " =" "3xx3^2" and "b^3" "=" "b^2xxb#
#4" "=2^2" and " a^3" "=a^2xxa#
'.........................................................................................

#" Write as "sqrt(2^2xxa^2xxa)/sqrt(3^2xx3xxb^2xxb)#

Take the squared values outside the square roots giving

#" "color(blue)((2a sqrt(a))/(3bsqrt(3b)))#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Now to 'get rid' of the square root in the denominator")#

Multiply by 1 and you do not change the value or how it looks.
1 can come in many forms: #1/1"; "(-1)/(-1)"; "2/2"; "sqrt(3b)/sqrt(3b)#

So we can multiply by 1 and not change the inherent value but we can change the way it looks.

#color(brown)("Multiply by 1 but in the form of "1=sqrt(3b)/sqrt(3b))#

#(2a sqrt(a))/(3bsqrt(3b))xxsqrt(3b)/sqrt(3b)" "=" "(2asqrt(a)sqrt(3b))/(3bsqrt(3b)sqrt(3b))#

But #sqrt(3b)xxsqrt(3b)=3b# giving:

#(2asqrt(a)sqrt(3b))/(3bxxb)" "=" "(2asqrt(a)sqrt(3b))/(3b^2)#

But #sqrt(a)sqrt(3b) = sqrt(3ab)#

#color(brown)("If you can take things out of square roots you can put things back in!")#

#" "color(green)((2asqrt(3ab))/(3b^2))#