How do you simplify #root3(4)/root5(8)#?

Question

Julia Adams · Accepted Answer

#root(3)(4)/root(5)(8) = root(15)(2)#

Since we are dealing with positive radicands, we can freely combine the exponents like this:

#root(3)(4)/root(5)(8) = (2^2)^(1/3) / (2^3)^(1/5)#

#color(white)(root(3)(4)/root(5)(8)) = 2^(2/3) / 2^(3/5)#

#color(white)(root(3)(4)/root(5)(8)) = 2^(2/3-3/5)#

#color(white)(root(3)(4)/root(5)(8)) = 2^(10/15-9/15)#

#color(white)(root(3)(4)/root(5)(8)) = 2^(1/15)#

#color(white)(root(3)(4)/root(5)(8)) = root(15)(2)#

Savannah Anderson · Answer

undefined To simplify the expression root3(4)/root5(8), we can rationalize the denominator by multiplying both the numerator and denominator by root5(8). This gives us root3(4) * root5(8) / root5(8) * root5(8). Simplifying further, we have root3(4 * 8) / root5(8 * 8). This simplifies to root3(32) / root5(64). Since 32 can be simplified to 16 * 2 and 64 can be simplified to 16 * 4, we have root3(16 * 2) / root5(16 * 4). Simplifying again, we get root3(16) * root3(2) / root5(16) * root5(4). The square root of 16 is 4, so we have 4 * root3(2) / 4 * root5(4). The 4s cancel out, leaving us with root3(2) / root5(4). Finally, we can simplify root5(4) to 2, giving us the final answer of root3(2) / 2.