Apply the properties of integer exponents to generate an equivalent algebraic expression to (4x4y−4)3(x7y3)2 .(1 point) Responses 64x20 Start Fraction 64 over x superscript 20 baseline End Fraction 4x2y18 Start Fraction 4 over x squared y superscript 18 baseline Endfraction 64x2y18 Start Fraction 64 over x squared y superscript 18 baseline Endfraction x2y1864
The correct equivalent algebraic expression is 64x^2y^18.
To generate an equivalent algebraic expression, we can apply the properties of integer exponents. Here's the step-by-step process:
Step 1: Expand the first expression (4x^4y^-4)^3:
(4^3)(x^4)^3(y^-4)^3
Step 2: Simplify the exponents inside the parentheses:
(64)(x^12)(y^-12)
Step 3: Expand the second expression (x^7y^3)^2:
(x^7)^2(y^3)^2
Step 4: Simplify the exponents inside the parentheses:
x^14y^6
Step 5: Combine the expanded expressions:
(64)(x^12)(y^-12)(x^14)(y^6)
Step 6: Simplify the expression by combining like terms:
64x^(12+14)y^(-12+6)
Step 7: Simplify the exponents:
64x^26y^-6
Therefore, the equivalent algebraic expression is 64x^26y^-6.
To generate an equivalent algebraic expression, we can use the properties of integer exponents. The properties that we will use are:
1. The Power of a Power Property: (a^m)^n = a^(m*n)
2. The Product of Powers Property: (ab)^n = a^n * b^n
3. The Quotient of Powers Property: (a/b)^n = a^n / b^n
Let's apply these properties step by step to simplify the given expression:
Step 1: Apply the Power of a Power Property to (4x^4y^-4)^3:
(4^3 * x^(4*3) * y^(-4*3))
Step 2: Simplify the exponents:
(64 * x^12 * y^-12)
Step 3: Apply the Product of Powers Property to (x^7y^3)^2:
(x^(7*2) * y^(3*2))
Step 4: Simplify the exponents:
(x^14 * y^6)
Step 5: Combine the results from Step 2 and Step 4:
(64 * x^12 * y^-12) * (x^14 * y^6)
Step 6: Multiply the coefficients and combine the variables with the same base:
(64 * x^(12+14) * y^(-12+6))
Step 7: Simplify the exponents:
(64 * x^26 * y^-6)
Step 8: Apply the Quotient of Powers Property to y^-6:
64x^26 / y^6
Therefore, the equivalent algebraic expression to (4x^4y^-4)^3 * (x^7y^3)^2 is 64x^26 / y^6.