Use the product property for radicals to simplify the following radical expressions as much as possible. Assume all variables represent positive numbers. Use sqrt(x) for √x and root(x)(y) for x√y.
3√750x^4y^7
To simplify the given expression using the product property for radicals, we can break down the radicand into its prime factorization:
750 = 2 * 3 * 5^2
x^4 = x * x * x * x
y^7 = y * y * y * y * y * y * y
Now, let's rearrange the factors within the radical as a product of square roots and cube roots:
3√(2 * 3 * 5^2 * x * x * x * x * y * y * y * y * y * y * y)
Next, we can group the common factors together and simplify:
3√(2 * (3 * 5 * 5) * (x * x * x * x) * (y * y * y * y * y * y * y))
Simplifying further:
3√(2 * 3 * 25 * x^4 * y^7)
Now, we can take the cube root of each factor that has a power of 3:
= 2√(3 * 25 * x^4 * y^6) * √(y)
Taking the square root of the remaining factors:
= 2√(3 * 5^2 * (x^2)^2 * (y^3)^2) * √y
Simplifying further gives:
= 2 * 5 * x^2 * y^3 * √(3y)
Therefore, the simplified expression is:
10x^2y^3√(3y)
To simplify the radical expression 3√750x^4y^7 using the product property for radicals, we need to break down the radicand (the expression under the radical sign) into prime factors.
Let's begin by finding the prime factorization of 750:
750 = 2 * 375
= 2 * 5 * 75
= 2 * 5 * 3 * 25
= 2 * 5 * 3 * 5 * 5
= 2 * 3 * 5^3
Now, let's rewrite the given radical expression with the prime factorization:
3√(2 * 3 * 5^3 * x^4 * y^7)
Next, we can separate the prime factors inside the radical sign based on their powers:
3√(2) * 3√(3) * 3√(5^3) * 3√(x^4) * 3√(y^7)
Simplifying each separate radical:
3√(2) * 3√(3) * 5 * √(x^4) * y^2 * √(y)
Since the square root of x^4 is x^2, we can simplify further:
3√(2) * 3√(3) * 5x^2 * y^2 * √(y)
Finally, we can multiply the coefficients and combine like terms:
15x^2y^2 * 3√(2) * 3√(3) * √(y)
Therefore, the simplified form of the radical expression 3√750x^4y^7 is 15x^2y^2 * 3√2 * 3√3 * √y.
Are you using ^ to indicate "to the power of"?
if so...
it appears you are asked to simplify
3 times the root of 750...
but then with no brackets it gets a wee bit muddy...
please advise.