Simplify the radical. Assume that all variables represent positive real numbers.
cube root (120m^20 n^2 / mn^14)
To simplify the given expression, we need to simplify the cube root of the given expression:
cube root (120m^20 n^2 / mn^14)
To simplify this expression, we will break down each term inside the cube root individually.
Step 1: Simplify the numerator (120m^20n^2):
- The cube root of 120 can be simplified by finding the prime factors. The prime factorization of 120 is 2^3 × 3 × 5.
- For the variable terms, we add the exponents: m^20 × n^2 = m^(20+2) = m^22.
- Therefore, the simplified numerator is 8m^22n^2.
Step 2: Simplify the denominator (mn^14):
- The denominator does not need further simplification.
Now, we can rewrite the expression as:
cube root (8m^22n^2 / mn^14)
Step 3: Cancel out the common factors:
- We can cancel out mn from the numerator and denominator, leaving us with:
cube root (8m^21n^-12).
Step 4: Simplify the remaining expression:
- The cube root of 8 is 2, as 2 × 2 × 2 = 8.
- For variable terms, we can write exponents in the form of fractions. So, m^21 = m^(7/3) and n^-12 = n^(-4).
Combining these simplifications, we have:
2m^(7/3)n^(-4)
Therefore, the simplified radical expression is 2m^(7/3)n^(-4).