Simplify the radical expression.
√63x^15y^9/7xy^11
To simplify this expression, we can first divide the coefficients (numbers outside the radical sign) and simplify the variables inside the radical sign using the rules of exponents:
√(63x^15y^9/7xy^11) = √(9x^4/y^2)
Next, we can split the radical sign using the product rule:
√(9x^4/y^2) = √9 * √(x^4/y^2)
The square root of 9 is 3, and we can simplify the expression inside the radical sign using the quotient rule:
√(x^4/y^2) = √(x^4)/√(y^2) = x^2/y
Putting it all together, we get:
√(63x^15y^9/7xy^11) = √9 * √(x^4/y^2) = 3 * x^2/y = 3x^2/y
To simplify the radical expression √(63x^15y^9)/(7xy^11), we can follow these steps:
Step 1: Rewrite the expression using fractional exponents.
√(63x^15y^9)/(7xy^11) can be rewritten as (63x^15y^9)^(1/2) / (7xy^11)^(1/2).
Step 2: Simplify the exponents.
The exponent 1/2 means taking the square root of the expression inside the parentheses.
So, (63x^15y^9)^(1/2) = √(63x^15y^9).
Similarly, (7xy^11)^(1/2) = √(7xy^11).
Step 3: Simplify the square roots.
√(63x^15y^9) can be rewritten as √(9 * 7 * 7 * x^14 * x * y^8 * y).
Taking any perfect squares out from under the square root, we get: 3 * 7 * x^7 * y^4 * √(x * y).
Similarly, √(7xy^11) can be simplified as √(7 * x * y^10) = √(7xy^2 * y^8) = √(7xy^2) * √(y^8) = √(7xy^2) * y^4.
Step 4: Simplify the expression.
Combining the simplified square roots, we have:
(3 * 7 * x^7 * y^4 * √(x * y)) / (√(7xy^2) * y^4).
We can cancel out the common factors in the numerator and denominator.
Cancelling out the y^4 terms, we get:
(3 * 7 * x^7 * y^4 * √(x * y)) / (y^4 * √(7xy^2)).
Finally, cancelling out the y^4 terms, we have:
(3 * 7 * x^7 * √(x * y)) / √(7xy^2).
Therefore, the simplified radical expression is:
(21x^7 * √(x * y)) / √(7xy^2).