Simplify the radical expression completely. DO NOT change to a decimal, give an exact answer. √ 72x^3y^8 / 2x^13y^ 5
Simplify the radical expression completely. DO NOT change to a decimal, give an exact answer. Show your work. √240x^12y^3 / 5xy^9
please help
sqrt(9*8)/2 * x^(12-1)y^(3-9)
the coefficent simplifies to sqrt(9*8)/2 = 3sqrt2
To simplify the radical expression √(72x^3y^8) / (2x^13y^5), first, let's break down the given expression into separate parts:
√(72x^3y^8) / (2x^13y^5)
Now, let's simplify the expression under the radical sign:
√(72x^3y^8) = √(72) √(x^3) √(y^8)
Next, let's simplify each individual square root:
√(72) = √(36 x 2) = 6 √2
√(x^3) = √(x^2 × x) = x√x
√(y^8) = √(y^4 × y^4) = y^4
Now, let's combine the simplified square roots:
√(72x^3y^8) = 6x√(2xy^4)
Finally, let's simplify the rest of the expression:
6x√(2xy^4) / (2x^13y^5)
The x in the numerator and the x in the denominator can be simplified and canceled out, leaving:
3√(2xy^4) / (x^12y^5)
So, the simplified radical expression is 3√(2xy^4) / (x^12y^5).
Now, let's simplify the second expression, √(240x^12y^3) / (5xy^9):
√(240x^12y^3) = √(16 x 15 x x^4 x x^4 x y^2 x y)
√(16) = 4
√(15) cannot be simplified further, so it remains as √15
√(x^4) = x^2
√(y^2) = y
Now, let's combine the simplified square roots:
√(240x^12y^3) = 4x^6y√15
Next, let's simplify the rest of the expression:
(4x^6y√15) / (5xy^9)
The x in the numerator and the x in the denominator can be simplified and canceled out, and the y in the numerator and the y in the denominator can be simplified and canceled out, leaving:
4x^5√15 / (5y^8)
So, the simplified radical expression is 4x^5√15 / (5y^8).
To simplify the radical expression √72x^3y^8 / 2x^13y^5, we can follow these steps:
Step 1: Simplify the numerator and denominator separately.
For the numerator, notice that 72 can be factored into its prime factors: 2 * 2 * 2 * 3 * 3. Since we are looking for perfect square factors, we can pair them up: 2 * 2 = 4 and 3 * 3 = 9. Thus, we can simplify the numerator to 4√(2x^3y^8).
For the denominator, we have 2x^13y^5.
Step 2: Simplify the fraction by dividing the square roots.
Now, we have the expression 4√(2x^3y^8) / 2x^13y^5, which simplifies to 2√(2x^3y^8) / x^13y^5.
Step 3: Simplify the radical expression.
We can simplify the radical by identifying perfect square factors. Notice that x^3 and y^8 are perfect square terms. Since the square root of x^3 is x^(3/2), and the square root of y^8 is y^(8/2) = y^4, we can rewrite the expression as (2x^(3/2)y^4√2) / x^13y^5.
Step 4: Simplify the expression further.
To simplify the expression further, we can combine like terms in the numerator. Notice that x^(3/2) / x^13 is equivalent to x^((3/2) - 13) = x^(-23/2). Similarly, y^4 / y^5 is equal to y^(4 - 5) = y^-1.
So, our final simplified expression is: (2√2x^(-23/2)y^(-1)).
Now, let's move on to the next expression √240x^12y^3 / 5xy^9 and simplify it.
Step 1: Simplify the numerator and denominator separately.
For the numerator, notice that 240 can be factored into its prime factors: 2 * 2 * 2 * 2 * 3 * 5. We can pair up perfect square factors: 2 * 2 = 4. So, the numerator simplifies to 4√(60x^12y^3).
For the denominator, we have 5xy^9.
Step 2: Simplify the fraction by dividing the square roots.
Now, we have the expression 4√(60x^12y^3) / 5xy^9, which simplifies to 4√(60x^12y^3) / 5xy^9.
Step 3: Simplify the radical expression.
We can simplify the radical by identifying perfect square factors. Notice that 60 can be factored into 2 * 2 * 3 * 5. Pairing up perfect squares, we have 2 * 2 = 4 and 5 is already a perfect square. So, the numerator simplifies to 4√(4 * 3 * 5 * x^12y^3).
Step 4: Simplify the expression further.
Breaking down the radical expression, 4√(4 * 3 * 5 * x^12y^3), we can simplify further. Notice that x^12 and y^3 are perfect square terms. The square root of x^12 is x^(12/2) = x^6, and the square root of y^3 is y^(3/2). Thus, the numerator becomes 4x^6y^(3/2)√(4 * 3 * 5).
Simplifying the remaining radical, √(4 * 3 * 5), becomes √(60) = √(4 * 15) = 2√(15).
So, our final simplified expression is: (4x^6y^(3/2)2√(15)) / 5xy^9, which simplifies to (8x^6y^(3/2)√15) / 5xy^9.