To generate equivalent expressions with only positive exponents, we can use the following properties of integer exponents:
1. (a^m)^n = a^(m*n)
2. a^(-m) = 1/(a^m)
3. a^m * a^n = a^(m+n)
4. (a/b)^m = a^m/b^m
Using these properties, let's simplify the given expression:
(4^3 ⋅ 7^(-2))^(-5) 7^9 ⋅ 4^(-13)
= (64 ⋅ 1/49)^(-5) 7^9 ⋅ 1/(4^13) (Using properties 2 and 4 to make the exponents positive)
= (64/49)^(-5) 7^9/4^13 (Simplifying the expression inside the parentheses)
= (49/64)^5 7^9/4^13 (Taking the reciprocal of 64/49 to make the exponent positive)
= 49^5/(64^5) 7^9/4^13 (Using property 3 to combine the two bases)
= (7^2)^5/(4^2)^5 7^9/4^13 (Writing 49 and 64 as powers of 7 and 4 respectively)
= 7^10/4^10 7^9/4^13 (Using property 1 to multiply the exponents)
= 7^(10+9)/4^(10+13) (Using property 3 to combine the exponents)
= 7^19/4^23 (Simplifying the exponents)
Therefore, the equivalent expression is 7^19/4^23.
Apply the Properties of Integer Exponents to generate equivalent expressions to (4^3 ⋅ 7^−2)^−5 7^9 ⋅ 4^−13 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
(4^3 ⋅ 7^−2)^−5 7^9 ⋅ 4^−13=?
1 answer