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)
To simplify the expression (4^3 ⋅ 7^−2)^−5 / 7^9 ⋅ 4^−13, we can apply the properties of integer exponents.
First, let's simplify the expression within the parentheses: (4^3 ⋅ 7^−2)^−5.
Using the product rule of exponents, we can simplify 4^3 ⋅ 7^−2 as (4 ⋅ 4 ⋅ 4) ⋅ (1 / 7^2), which simplifies to 64 ⋅ (1 / 49).
Therefore, (4^3 ⋅ 7^−2)^−5 becomes (64 / 49)^−5.
Now let's simplify the expression outside the parentheses: 7^9 ⋅ 4^−13.
Using the power rule of exponents, we can simplify 7^9 ⋅ 4^−13 as 7^9 ⋅ (1 / 4^13).
Therefore, (4^3 ⋅ 7^−2)^−5 / 7^9 ⋅ 4^−13 becomes (64 / 49)^−5 / 7^9 ⋅ (1 / 4^13).
Now, let's simplify further:
(64 / 49)^−5 is the same as (49 / 64)^5.
Similarly, 1 / 4^13 can be written as 1 / (2^2)^13, which simplifies to 1 / 2^26.
Therefore, the expression becomes (49 / 64)^5 / 7^9 ⋅ 1 / 2^26.
Now, let's simplify the final expression:
(49 / 64)^5 is the same as 49^5 / 64^5.
Therefore, the expression becomes 49^5 / 64^5 / 7^9 ⋅ 1 / 2^26.
Now, let's calculate the values:
49^5 = 49 * 49 * 49 * 49 * 49 = 5764801.
64^5 = 64 * 64 * 64 * 64 * 64 = 1073741824.
7^9 = 7 * 7 * 7 * 7 * 7 * 7 * 7 * 7 * 7 = 40353607.
2^26 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 67108864.
Substituting these values, the expression becomes:
5764801 / 1073741824 / 40353607 * 1 / 67108864.
Simplifying further, the expression becomes:
5764801 / (1073741824 * 40353607 * 67108864).
Calculating the product in the denominator, we get:
1073741824 * 40353607 * 67108864 = 28925465497693824.
Therefore, the expression simplifies to:
5764801 / 28925465497693824.
This is the final simplified fraction with no remaining exponents.
To generate equivalent expressions with only positive exponents, we can apply the properties of integer exponents.
First, let's simplify the expression using the properties of exponents:
(4^3 ⋅ 7^−2)^−5 / 7^9 ⋅ 4^−13
= (4^(3 * -5) ⋅ 7^(-2 * -5)) / 7^9 ⋅ 4^(-13)
= 4^(-15) ⋅ 7^(10) / 7^9 ⋅ 4^(-13)
Now, let's apply the property of exponents where a^m / a^n = a^(m-n):
= 4^(-15 - (-13)) ⋅ 7^(10 - 9)
= 4^(-15 + 13) ⋅ 7^(1)
= 4^(-2) ⋅ 7
Next, let's apply the property of exponents where a^(-n) = 1 / a^n to make the exponent positive:
= 1 / 4^2 ⋅ 7
= 1 / 16 ⋅ 7
= 7 / 16
So, the simplified fraction without remaining exponents is 7/16.