(4^3 ⋅ 7^−2)^−5 / 7^9 ⋅ 4^−13
To simplify the expression, we'll first deal with the exponents:
(4^3 ⋅ 7^(-2))^(-5) = 4^(3 * -5) ⋅ 7^(-2 * -5)
= 4^(-15) ⋅ 7^(10)
= 7^(10) / 4^(15)
Now we can rewrite the entire expression:
(7^(10) / 4^(15)) / (7^9 ⋅ 4^(-13))
When dividing fractions, we can multiply by the reciprocal of the second fraction instead:
(7^(10) / 4^(15)) * (1 / (7^9 ⋅ 4^(-13)))
Now we can simplify the expression by canceling out common factors:
= 7^(10 - 9) / 4^(15 + 13)
= 7^(1) / 4^(28)
= 7 / 4^28
Therefore, the simplified expression is 7 / 4^28.
To simplify the given expression, let's break it down step-by-step:
Step 1: Simplify the expression inside the parentheses.
Inside the parentheses, we have (4^3 ⋅ 7^−2). Let's simplify this first:
4^3 = 4 x 4 x 4 = 64
7^−2 = 1 / (7^2) = 1 / (7 x 7) = 1 / 49
So, (4^3 ⋅ 7^−2) = 64 x (1 / 49) = 64 / 49.
Step 2: Simplify the term (4^3 ⋅ 7^−2)^−5.
Recall that raising a term to a negative exponent is the same as taking the reciprocal of the term raised to the positive exponent. Therefore, (4^3 ⋅ 7^−2)^−5 = 1 / [(4^3 ⋅ 7^−2)^5].
Since we already simplified (4^3 ⋅ 7^−2) in Step 1, we can substitute this value into the expression:
(4^3 ⋅ 7^−2)^−5 = 1 / [(64 / 49)^5]
Step 3: Simplify the term 7^9.
7^9 = 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 = 40353607.
Step 4: Simplify the term 4^−13.
4^−13 = 1 / (4^13) = 1 / (4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4)
= 1 / (67108864) ≈ 1.4901 x 10^−8.
Step 5: Substitute the simplified values into the expression.
(4^3 ⋅ 7^−2)^−5 / 7^9 ⋅ 4^−13 = (1 / [(64 / 49)^5]) / (40353607 x 1.4901 x 10^−8).
Step 6: Simplify further if necessary.
To simplify further, we can rewrite the division of fractions as multiplication by the reciprocal:
(1 / [(64 / 49)^5]) / (40353607 x 1.4901 x 10^−8) = (1 / [(64 / 49)^5]) x (1 / (40353607 x 1.4901 x 10^−8))
= 1 / [(64 / 49)^5 x (40353607 x 1.4901 x 10^−8)].
At this point, we cannot simplify further without knowing if there are any specific values assigned to 10.