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.(1 point)

Question

(4^3 ⋅ 7^−2)^−5 / 7^9 ⋅ 4^−13 = ________

Answer 1

To generate equivalent expressions with positive exponents, we can use the following properties of integer exponents:

1. For any nonzero number a, a^(-n) = 1/a^n
2. (a^m)^n = a^(m*n)
3. a^n * a^m = a^(n+m)
4. a^0 = 1

Let's simplify the expression step by step:

(4^3 ⋅ 7^−2)^−5 / 7^9 ⋅ 4^−13

Using property 2, we can simplify the numerator:

= (4^(-15) ⋅ 7^10) / 7^9 ⋅ 4^(-13)

Using properties 1 and 3, we can simplify further:

= (7^10 / 4^15) / 7^9 ⋅ 4^(-13)

Using properties 1 and 3 again, we can simplify the denominator:

= (7^10 / 4^15) / 7^9 / 4^13

Now, let's use property 3 to combine the fractions:

= (7^10 / 4^15) * (7^(-9) / 4^13)

Using property 3 again, let's combine the bases:

= 7^(10-9) / 4^(15+13)

= 7^1 / 4^28

Now, we can simplify the expression by evaluating the exponents:

= 7 / (2^56)

Therefore, the simplified fraction is 7 / 2^56.

Answer 2

WRONG i need the correct answer

Answer 3

Apologies for the mistake. Let's re-evaluate the expression using the properties of integer exponents correctly:

(4^3 ⋅ 7^−2)^−5 / 7^9 ⋅ 4^−13

Using the property (a^m)^n = a^(m*n), we can simplify the numerator:

= (4^(3*-5) ⋅ 7^(-2*-5)) / 7^9 ⋅ 4^(-13)

= 4^(-15) ⋅ 7^10 / 7^9 ⋅ 4^(-13)

Using the property a^n * a^m = a^(n+m), we can combine the like bases:

= 4^(-15 + (-13)) ⋅ 7^(10 - 9)

= 4^(-28) ⋅ 7^1

Using the property a^0 = 1, we simplify further:

= 4^(-28) ⋅ 7

Now, since we want the answer in positive exponents, we can utilize the property a^(-n) = 1/a^n to convert the negative exponent:

= 1/4^28 ⋅ 7

= 7/4^28

Therefore, the equivalent expression with positive exponents is 7/4^28.