Apply the properties of integer exponents to simplify the algebraic expression (2xy^3)^4⋅(x^2 y)^3.
Responses
16x^10 y^15
16x^25
32x^10 y^11
x^10 y^15
To simplify the expression (2xy^3)^4⋅(x^2 y)^3 using the properties of integer exponents, we can simplify each base individually and then multiply the results.
First, let's simplify the base (2xy^3)^4:
Using the power of a product property, we can raise each factor inside the parentheses to the 4th power.
(2xy^3)^4 = 2^4 * (x^1)^4 * (y^3)^4 = 16x^4 * x^4 * y^12 = 16x^8y^12
Next, let's simplify the base (x^2 y)^3:
Using the power of a product property, we can raise each factor inside the parentheses to the 3rd power.
(x^2 y)^3 = (x^2)^3 * (y^1)^3 = x^6 * y^3
Finally, let's multiply the two simplified bases together:
16x^8y^12 * x^6 * y^3 = 16x^8 * x^6 * y^12 * y^3 = 16x^14 * y^15
The simplified algebraic expression is 16x^14 * y^15. So, the correct response is:
16x^14 y^15
To simplify the expression (2xy^3)^4 ⋅ (x^2y)^3 using the properties of integer exponents, we need to apply the power of a power rule and the product of powers rule.
First, let's simplify (2xy^3)^4:
(2xy^3)^4 can be expanded as (2^4)(x^4)(y^12) since the exponent 4 is applied to each term inside the parentheses.
Simplifying further, we have 16x^4y^12.
Next, let's simplify (x^2y)^3:
(x^2y)^3 can be expanded as (x^2)^3(y^3)^3 using the power of a power rule.
Simplifying further, we have (x^6)(y^9).
Now, we can multiply the simplified expressions:
(16x^4y^12) ⋅ (x^6)(y^9) can be simplified by multiplying the coefficients and combining the variables:
16x^4 ⋅ x^6 ⋅ y^12 ⋅ y^9 = 16x^(4+6) ⋅ y^(12+9) = 16x^10 ⋅ y^21.
Therefore, the simplified form of the expression (2xy^3)^4 ⋅ (x^2y)^3 is 16x^10y^21.