SOLUTION: (36^2a^3/b^2)^1/4(8a^3/b^-2)^1/3

Question 391288: (36^2a^3/b^2)^1/4(8a^3/b^-2)^1/3
Answer by haileytucki(390) (Show Source): You can put this solution on YOUR website!
((36^(2)a^(3))/(b^(2)))^((1)/(4))*((8a^(3))/(b^(-2)))^((1)/(3))
Squaring a number is the same as multiplying the number by itself (36*36). In this case, 36 squared is 1296.
((1296a^(3))/(b^(2)))^((1)/(4))*((8a^(3))/(b^(-2)))^((1)/(3))
Expand the exponent (((1)/(4))) to the expression.
(1296^((1)/(4))a^(3*((1)/(4))))/((b^(2))^((1)/(4)))*((8a^(3))/(b^(-2)))^((1)/(3))
Expand the exponent (((1)/(4))) to the expression.
(1296^((1)/(4))a^(3*((1)/(4))))/(b^(2*((1)/(4))))*((8a^(3))/(b^(-2)))^((1)/(3))
Multiply 3 by each term inside the parentheses.
(1296^((1)/(4))a^((3)/(4)))/(b^(2*((1)/(4))))*((8a^(3))/(b^(-2)))^((1)/(3))
Multiply 2 by each term inside the parentheses.
(1296^((1)/(4))a^((3)/(4)))/(b^((1)/(2)))*((8a^(3))/(b^(-2)))^((1)/(3))
An expression with a fractional exponent can be written as a radical with an index equal to the denominator of the exponent.
((~4:(1296))*a^((3)/(4)))/(b^((1)/(2)))*((8a^(3))/(b^(-2)))^((1)/(3))
Pull all perfect 4th roots out from under the radical. In this case, remove the 6 because it is a perfect 4th.
((6)*a^((3)/(4)))/(b^((1)/(2)))*((8a^(3))/(b^(-2)))^((1)/(3))
Multiply (6) by a^((3)/(4)) to get a^((3)/(4))(6).
(a^((3)/(4))(6))/(b^((1)/(2)))*((8a^(3))/(b^(-2)))^((1)/(3))
Multiply a^((3)/(4)) by each term inside the parentheses.
(6a^((3)/(4)))/(b^((1)/(2)))*((8a^(3))/(b^(-2)))^((1)/(3))
Move all negative exponents from the denominator to the numerator and make the exponents positive. A negative exponent follows the rule: a^(-n)=(1)/(a^(n)).
(6a^((3)/(4)))/(b^((1)/(2)))*(8a^(3)(b^(2)))^((1)/(3))
Multiply out the values in the numerator and denominator to remove the parentheses in the expression.
(6a^((3)/(4)))/(b^((1)/(2)))*(8a^(3)b^(2))^((1)/(3))
Expand the exponent (((1)/(3))) to the expression.
(6a^((3)/(4)))/(b^((1)/(2)))*8^((1)/(3))a^(3*((1)/(3)))b^(2*((1)/(3)))
Multiply 3 by each term inside the parentheses.
(6a^((3)/(4)))/(b^((1)/(2)))*8^((1)/(3))ab^(2*((1)/(3)))
Multiply 2 by each term inside the parentheses.
(6a^((3)/(4)))/(b^((1)/(2)))*8^((1)/(3))ab^((2)/(3))
An expression with a fractional exponent can be written as a radical with an index equal to the denominator of the exponent.
(6a^((3)/(4)))/(b^((1)/(2)))*((~3:(8))*ab^((2)/(3)))
Pull all perfect cube roots out from under the radical. In this case, remove the 2 because it is a perfect cube.
(6a^((3)/(4)))/(b^((1)/(2)))*((2)*ab^((2)/(3)))
Multiply (2) by ab^((2)/(3)) to get ab^((2)/(3))(2).
(6a^((3)/(4)))/(b^((1)/(2)))*(ab^((2)/(3))(2))
Multiply ab^((2)/(3)) by each term inside the parentheses.
(6a^((3)/(4)))/(b^((1)/(2)))*2ab^((2)/(3))
Multiply (6a^((3)/(4)))/(b^((1)/(2))) by 2ab^((2)/(3)) to get (12a^((7)/(4))b^((2)/(3)))/(b^((1)/(2))).
(12a^((7)/(4))b^((2)/(3)))/(b^((1)/(2)))
Reduce the exponents of b by subtracting the denominator exponents from the numerator exponents.
12a^((7)/(4))b^(((2)/(3))-((1)/(2)))
Multiply -1 by each term inside the parentheses.
12a^((7)/(4))b^((2)/(3)-(1)/(2))
Combine -(1)/(2)+(2)/(3) into a single expression by finding the least common denominator (LCD). The LCD of -(1)/(2)+(2)/(3) is 6.
12a^((7)/(4))b^((1)/(6))
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