Apply the Properties of Integer Exponents to generate equivalent expressions to (7^−3)^7 ⋅ 7^20/6^−2 with only positive exponents, then solve the expression. Provide the answer as an improper fraction.(1 point)
To simplify the expression and apply the properties of integer exponents, we start by raising the base of the first term, 7, to the power outside the parentheses, which is -3 for a negative exponent:
(7^(-3))^7 = 7^(-3*7) = 7^(-21)
Then, we can multiply this expression by the second term, 7^20, using the property of multiplying exponents with the same base:
7^(-21) * 7^20 = 7^(-21+20) = 7^(-1)
Finally, we can divide this expression by 6^(-2) by using the property of dividing exponents with the same base:
7^(-1) / 6^(-2) = 7^(-1) * 6^2 = 7^(-1) * 36 = 36/7
So the simplified expression is 36/7, which can be written as an improper fraction.
Simplify the expression (4x^3)^2 using the properties of integer exponents.(1 point)
16x^6
16 x superscript 6 baseline
16x^5
16 x superscript 5 baseline
4x^6
4 x superscript 6 baseline
8x^6
The expression (4x^3)^2 can be simplified using the property of raising a power to a power, which states that (a^m)^n = a^(m*n). In this case, a is 4x^3, m is 2, and n is 1.
So, applying the property:
(4x^3)^2 = 4^(2*1) * (x^3)^(2*1) = 4^2 * x^(3*2) = 16 * x^6
Therefore, the simplified expression is 16x^6.
Apply the properties of integer exponents to generate an equivalent algebraic expression to (4x^4y^−4)^3/(x^7y^3)^2 .(1 point)
Responses
x^2y^18/64
Start Fraction x squared y superscript 18 baseline over 64 End Fraction
4/x^2y^18
Start Fraction 4 over x squared y superscript 18 baseline End Fraction
64/x^20
Start Fraction 64 over x superscript 20 baseline End Fraction
64/x^2y^18
To generate an equivalent algebraic expression to (4x^4y^−4)^3/(x^7y^3)^2, we can apply the properties of integer exponents.
First, we raise each term within the parentheses to the power outside of it:
(4x^4y^−4)^3 = 4^3 * (x^4)^3 * (y^−4)^3 = 64x^12y^−12
(x^7y^3)^2 = (x^7)^2 * (y^3)^2 = x^14y^6
Next, we divide the two expressions using the property of dividing exponents with the same base:
(64x^12y^−12)/(x^14y^6) = 64/(x^14/x^12) * (x^12/y^6)/(y^12/y^6)
Simplifying further, we get:
64 * x^(12-14) * (x^12/y^6) * (y^6/y^12) = 64 * x^(-2) * (x^12/y^6) * (1/y^6)
Combining like terms and simplifying, we have:
64 * (x^12 * x^(-2))/(y^6 * y^6) = 64 * x^10/y^12
Therefore, the equivalent algebraic expression is 64x^10/y^12.