(4^3 ⋅ 7^−2)^−5 / 7^9 ⋅ 4^−13 as a fraction
First, let's simplify the numerator:
(4^3 * 7^(-2))^(-5) is equal to (64 * (1/49))^(-5) which simplifies to (64/49)^(-5).
Next, let's simplify the denominator:
7^9 * 4^(-13) is equal to 7^9 * (1/4^13) which simplifies to 7^9 / 4^13.
Now, we can rewrite the expression as:
(64/49)^(-5) / (7^9 / 4^13)
To divide by a fraction, we can multiply by the reciprocal, so the expression becomes:
(64/49)^(-5) * (4^13 / 7^9)
Now, we can simplify each part separately:
(64/49)^(-5) = (49/64)^5
(49/64)^5 = (49^5 / 64^5)
Next,
(4^13 / 7^9) = (2^26 / 7^9)
Now, combining the simplified parts, we get:
(49^5 / 64^5) * (2^26 / 7^9)
To combine the fractions, we can multiply the numerators and multiply the denominators:
(49^5 * 2^26) / (64^5 * 7^9)
Therefore, the expression (4^3 ⋅ 7^(-2))^(-5) / (7^9 ⋅ 4^(-13)) simplified as a fraction is:
(49^5 * 2^26) / (64^5 * 7^9)
(49^5 * 2^26) / (64^5 * 7^9) simplified
To simplify the expression (49^5 * 2^26) / (64^5 * 7^9), we can further simplify the numerator and denominator.
In the numerator:
49^5 is equal to (7^2)^5, so we can rewrite it as 7^10.
Similarly, 2^26 can be written as (2^6)^4, which simplifies to 64^4.
In the denominator:
64^5 is equal to (2^6)^5, so it simplifies to 2^30.
And 7^9 remains as it is.
Now, we have:
(7^10 * 64^4) / (2^30 * 7^9)
Since the bases are the same in both the numerator and denominator, we can subtract the exponents:
= (7^10 * 2^18) / (2^30 * 7^9)
Next, let's simplify the powers of 7:
= (7^(10-9) * 2^18) / (2^30)
= (7 * 2^18) / (2^30)
Now, we can simplify the powers of 2:
= (7 * 2^(18-30))
= (7 * 2^(-12))
Therefore, the expression (49^5 * 2^26) / (64^5 * 7^9) simplified as a fraction is:
(7 * 2^(-12))
To simplify the expression, let's break it down step by step:
1. First, simplify the numerator (4^3 ⋅ 7^−2)^−5.
- The exponent −5 indicates that we need to take the reciprocal of the expression to the fifth power.
- The reciprocal of (4^3 ⋅ 7^−2) is 1 / (4^3 ⋅ 7^−2).
- Applying the fifth power to the reciprocal, we get (1 / (4^3 ⋅ 7^−2))^5.
2. Next, simplify the denominator 7^9 ⋅ 4^−13.
3. Now, let's simplify the numerator and denominator separately.
Simplify the numerator:
- In the numerator, expand the exponents: 1 / (4^3 ⋅ 7^−2) = 1 / (4^3) ⋅ 1 / (7^−2).
- Simplify the exponents: 1 / 4^3 = 1 / 64, and 1 / 7^−2 = 1 / (1 / 7^2) = 7^2 = 49.
- Therefore, the numerator simplifies to 1 / 64 ⋅ 49.
Simplify the denominator:
- In the denominator, combine the exponents: 7^9 ⋅ 4^−13 = 7^9 / 4^13.
- Now simplify the exponents: 4^13 = 4^-13 * 4^26 = 4^26 / 4^13 = 4^13.
- Therefore, the denominator simplifies to 7^9 / 4^13.
4. Now that we have simplified the numerator and denominator, let's rewrite the expression in simplified form:
- (1 / 64 ⋅ 49) / (7^9 / 4^13).
5. To divide fractions, we multiply the first fraction by the reciprocal of the second fraction:
- So, (1 / 64 ⋅ 49) * (4^13 / 7^9).
- Combine the numerators and denominators: (1 ⋅ 4^13 ⋅ 49) / (64 ⋅ 7^9).
6. Finally, we have (4^13 ⋅ 49) / (64 ⋅ 7^9) as the simplified form of the expression.