(4^3 ⋅ 7^−2)^−5 / 7^9 ⋅ 4^−13 simplified fraction with no exponents
To simplify the expression (4^3 ⋅ 7^−2)^−5 / 7^9 ⋅ 4^−13, we can start by simplifying the numerator and denominator separately.
Numerator:
(4^3 ⋅ 7^−2)^−5
Using the property (a⋅b)^n = a^n ⋅ b^n, we can raise both the 4 and 7 to the power of -5:
= 4^(-5⋅3) ⋅ 7^(-5⋅(-2))
= 4^(-15) ⋅ 7^10
Denominator:
7^9 ⋅ 4^−13
Again using the property (a⋅b)^n = a^n ⋅ b^n, we can raise both the 4 and 7 to the powers of -13 and 9, respectively:
= 7^9 ⋅ 4^(-13)
= 7^9 ⋅ (1/4^13) (since 4^(-13) = 1/4^13)
Now we can simplify the fraction by dividing the numerator by denominator:
(4^(-15) ⋅ 7^10) / (7^9 ⋅ (1/4^13))
Using the properties of exponents, we can simplify further by subtracting the exponents when dividing with the same base:
= 4^(-15-13) ⋅ 7^(10-9)
= 4^(-28) ⋅ 7^1
Now we have a simplified fraction without exponents.
To simplify the given expression, we can apply the exponent rules. Let's break it down step by step:
Step 1: Simplify the exponents in the numerator.
(4^3 ⋅ 7^−2)^−5 = (64 ⋅ 1/49)^−5
Step 2: Simplify the exponents in the denominator.
7^9 ⋅ 4^−13 = (7^9)/(4^13)
Step 3: Simplify the exponents in the numerator and denominator.
(64 ⋅ 1/49)^−5 / (7^9)/(4^13) = (64^−5 ⋅ (1/49)^−5) / (7^9)/(4^13)
Step 4: Apply the exponent rules for negative exponents.
(64^−5 ⋅ (1/49)^−5) / (7^9)/(4^13) = (1/64^5 ⋅ 49^5) / (7^9)/(4^13)
Step 5: Apply the exponent rules for division.
(1/64^5 ⋅ 49^5) / (7^9)/(4^13) = (1/64^5 ⋅ 49^5) ⋅ (4^13/7^9)
Step 6: Simplify each term separately.
(1/64^5 ⋅ 49^5) ⋅ (4^13/7^9) = (1/1024^2 ⋅ 7^5) ⋅ (7^5/7^9 ⋅ 4^13)
Step 7: Apply the exponent rules for multiplication and division.
(1/1024^2 ⋅ 7^5) ⋅ (7^5/7^9 ⋅ 4^13) = 1/1024^2 ⋅ 7^5 ⋅ 7^5/7^9 ⋅ 4^13
Step 8: Apply the exponent rule for combining like bases.
1/1024^2 ⋅ 7^5 ⋅ 7^5/7^9 ⋅ 4^13 = 1/1024^2 ⋅ 7^(5+5)/7^9 ⋅ 4^13
Step 9: Simplify the exponents.
1/1024^2 ⋅ 7^(5+5)/7^9 ⋅ 4^13 = 1/1024^2 ⋅ 7^10/7^9 ⋅ 4^13
Step 10: Apply the exponent rule for division of like bases.
1/1024^2 ⋅ 7^10/7^9 ⋅ 4^13 = 1/1024^2 ⋅ 7^(10-9) ⋅ 4^13
Step 11: Simplify the exponents.
1/1024^2 ⋅ 7^(10-9) ⋅ 4^13 = 1/1024^2 ⋅ 7^1 ⋅ 4^13
Step 12: Combine any like terms.
1/1024^2 ⋅ 7^1 ⋅ 4^13 = 1/1024^2 ⋅ 7 ⋅ 4^13
Therefore, the simplified fraction with no exponents is: 1/1024^2 ⋅ 7 ⋅ 4^13.