Multiply: (4xy^2)^1/5 and (8x^2y)^1/5
The rule coming into play here is:
(a^x)(b^x) = (ab)^x
apply it, and let me know what you get
(4xy^2)^1/5 * (8x^2y)^1/5 = (4xy^2 * 8x^2y)^1/5 = (32x^3y^3)^1/5 =
(2^5x^3y^3)^1/5 = 2(x^3*y^3)^1/5 = 2(x^3/5y^3/5).
To multiply the two expressions:
Step 1: We can simplify the expressions inside the brackets by applying the exponent to each term.
For the first expression, (4xy^2)^(1/5), apply the exponent to each term:
= 4^(1/5) * x^(1/5) * (y^2)^(1/5)
For the second expression, (8x^2y)^(1/5), apply the exponent to each term:
= 8^(1/5) * (x^2)^(1/5) * y^(1/5)
Step 2: Simplify further by evaluating the roots:
= 4^(1/5) * x^(1/5) * y^(2/5)
= 8^(1/5) * x^(2/5) * y^(1/5)
Step 3: Finally, multiply the simplified expressions:
= (4^(1/5) * x^(1/5) * y^(2/5)) * (8^(1/5) * x^(2/5) * y^(1/5))
To multiply the numbers, multiply their coefficients:
= (2 * 2^(1/5) * x^(1/5 + 2/5) * y^(2/5 + 1/5))
Combine the exponents for x and y:
= 2 * 2^(1/5) * x^(3/5) * y^(3/5)
Therefore, the product of (4xy^2)^(1/5) and (8x^2y)^(1/5) is 2 * 2^(1/5) * x^(3/5) * y^(3/5).
To multiply expressions with the same base but different exponents, you can add the exponents together. In this case, we have two expressions with a base of x and y, and the exponents are both 1/5.
Let's first focus on the x term:
(4xy^2)^(1/5) = (4x)^(1/5) * (y^2)^(1/5)
The exponent on (4x) is 1/5, so we take the fifth root of 4x, which gives us (4x)^(1/5) = (4^(1/5)) * (x^(1/5)) = (2)(x^(1/5))
Now let's look at the y term:
(8x^2y)^(1/5) = (8x^2)^(1/5) * (y)^(1/5)
The exponent on (8x^2) is also 1/5, so we take the fifth root of 8x^2, which gives us (8x^2)^(1/5) = (8^(1/5)) * (x^(2/5)) = (2)(x^(2/5))
Finally, we multiply the x and y terms together:
(2)(x^(1/5)) * (2)(x^(2/5)) * (y)^(1/5) =
4 * x^((1/5) + (2/5)) * y^(1/5) =
4 * x^(3/5) * y^(1/5)
So, the product of (4xy^2)^(1/5) and (8x^2y)^(1/5) is 4 * x^(3/5) * y^(1/5).