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

Question

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

Roman Alvarez · Accepted Answer

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

Luna Angel · Answer

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

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

Benjamin Altschul · Answer

undefined To simplify $\sqrt{\frac{4a^3}{27b^3}}$, you can follow these steps:

1. Rewrite the expression under the square root sign as a fraction: $\frac{4a^3}{27b^3}$.
2. Factor the numerator and denominator to simplify: $4a^3 = 2^2 \cdot a^3$ and $27b^3 = 3^3 \cdot b^3$.
3. Take the square root of the factors: $\sqrt{2^2 \cdot a^3} = 2a \sqrt{a}$ and $\sqrt{3^3 \cdot b^3} = 3b \sqrt{b}$.
4. Combine the square roots: $\sqrt{\frac{4a^3}{27b^3}} = \frac{2a\sqrt{a}}{3b\sqrt{b}}$.
5. Simplify the expression by dividing the numerator and denominator by the greatest common factor (GCF), which is 2: $\frac{2a\sqrt{a}}{3b\sqrt{b}} = \frac{a\sqrt{a}}{\frac{3}{2}b\sqrt{b}}$.

Therefore, the simplified expression is $\frac{a\sqrt{a}}{\frac{3}{2}b\sqrt{b}}$.