Question

write (x^3y)1/4 in radical form

Answer 1

The trick to use here is the fact that all rational exponents have an equivalent form using square roots, cube roots, and in general: nth roots (for some positive integer n). We can convert between the two notations using the equation:

x^(m/n) = (n√)(x^m)

where the symbol (n√) means "nth root" (because my keyboard only has so many symbols). For example:

x^(3/2) = √(x³)
x^(5/3) = ∛(x⁵)
x^(3/4) = ∜(x³)

Furthermore, we can simplify the first two radical expressions by taking out a pair and trio, respectively, out of the radical:

x^(3/2) = √(x³) = x * √x
x^(5/3) = ∛(x⁵) = x * ∛x

So let's take a look at your problems. For the first problem, we can use the radical-form / rational-exponent equation almost directly:

(x^3y)^(1/4) = ∜(x³*y)

Since none of the exponents are greater than or equal to 4, the expression is now simplified completely. However, the next problem requires some simplification. So first we'll convert it to radical form:

x^(9/4) = ∜(x⁹) = ∜((x⁴)(x⁴)(x)) = x² * ∜x

where the last equation works because we can take out two quartets of x from the radical. That is, x⁸ on the inside is the same as x^(8/4) = x² on the outside.

Now we'll convert both terms to radical form and multiply them together to write the expression using a single radical:

x^(2/3)y^(5/3) = ∛(x²)*∛(y⁵) = ∛(x²y⁵) = y*∛(x²y²)

Finally, for the last problem, we can simplify and find common denominators for the exponents before converting to radical form:

a^(1/2)b^(4/3)c^(3/4)
= a^(6/12)*b^(16/12)*c^(9/12)
= a^(6/12)*b^(1+3/12)*c^(9/12)
= b*(a^(6/12)*b^(3/12)*c^(9/12))
= b*(12√)(a⁶b³c⁹)

where (12√) means "the 12th root," as you may recall.

Answer 2

thank you so much!

Answer 3

Welcome! (:

Answer 4

Do you need help with anything else? (:

Answer 5

im sure i will! lol im not the best at math! when i do ill post it! thanks!

Answer 6

Welcome