How do you simplify #root3(y^6)/(root3(27y)*root3(y^11))#?

Question

Abigail Allen · Accepted Answer

#1/(3y^2)# Because all the terms are cube rooted (and the expression contains no addition or subtraction), the cube root can be moved to the outside of the expression: #root3(y^6/(27y.y^11))# We can simplify the denominator using the fact that #a^n.a^m=a^(n+m)# Both terms in the denominator are #y# raised to a power, so when we multiply them we add the indices to get: #root3(y^6/(27y^12))# We can now divide the top and the bottom of the fraction by #y^6# giving us #1# in the numerator, as #y^6/y^6=1# For the denominator we use #a^n/a^m=a^(n-m)# to get: #27y^12/y^6=27y^(12-6)=27y^6# So the expression becomes: #root3(1/(27y^6))# The cube root of #27# is #3#. #root3(x)# is equivalent to #x^(1/3)#, so #root3(y^6)=y^(6/3)=y^2# Therefore the expression fully simplifies to: #1/(3y^2)#

Kennedy Adams · Answer

undefined To simplify the expression (root3(y^6))/(root3(27y)*root3(y^11)), we can start by simplifying the individual terms inside the roots.

First, let's simplify root3(y^6). Since the index of the root is 3, we can rewrite y^6 as (y^2)^3. Therefore, root3(y^6) simplifies to y^2.

Next, let's simplify root3(27y). We can rewrite 27 as 3^3, and since the index of the root is 3, we can rewrite root3(27y) as root3(3^3 * y) which simplifies to 3y.

Finally, let's simplify root3(y^11). Since the index of the root is 3, we can rewrite y^11 as (y^3)^3 * y^2. Therefore, root3(y^11) simplifies to y^3 * y^2, which is y^5.

Now, we can substitute these simplified terms back into the original expression:

(y^2) / (3y * y^5)

To simplify further, we can divide y^2 by y^5, which gives us 1/y^3. Therefore, the simplified expression is:

1 / (3y^3)