Apply the properties of integer exponents to generate equivalent expressions to (4^3 7^-2)^-5 over 7^9 / 4^-13 with only positive exponents then solve the expression, the answer will be a simplified fraction with no remaining exponents
the answer is 7 / 16
Let's break down the given expression and simplify it step by step.
1. (4^3 7^-2)^-5:
By applying the property of exponents, we can multiply the exponents within the parentheses:
(4^3 7^-2)^-5 = 4^(3*-5) 7^(-2*-5) = 4^(-15) 7^10
2. 7^9 / 4^-13:
To simplify this expression, we can change the negative exponent to a positive exponent by applying the property of exponents:
7^9 / 4^-13 = 7^9 / (1/4^13) = 7^9 4^13
Now, let's combine the two expressions:
(4^3 7^-2)^-5 / (7^9 / 4^-13) = (4^(-15) 7^10) / (7^9 4^13)
Using the property of exponents, we can divide the exponents with the same base inside the parentheses:
(4^(-15) 7^10) / (7^9 4^13) = 4^(-15-13) 7^(10-9) = 4^-28 7^1
Finally, we simplify the expression:
4^-28 7^1 = 1/4^28 * 7 = 7/4^28
The simplified fraction is 7/4^28.
Let's break down the problem step by step:
Step 1: Apply the exponent property to the expression (4^3 7^-2)^-5.
Using the rule (a^m)^n = a^(m*n), we can simplify the expression as follows:
(4^3 7^-2)^-5 = 4^(3*(-5)) 7^(-2*(-5)) = 4^(-15) 7^10
Step 2: Apply the exponent property to the expression 7^9 / 4^-13.
Using the rule a^m / a^n = a^(m-n), we can simplify the expression as follows:
7^9 / 4^-13 = 7^9 * 4^13
Step 3: Combine the two simplified expressions.
Now we can rewrite the combined expression as:
(4^(-15) 7^10) / (7^9 * 4^13)
Step 4: Rearrange the expression to have only positive exponents.
To achieve positive exponents, we can move the negative exponents from the numerator to the denominator by reciprocating the base:
(7^10 * 4^(-15)) / (7^9 * 4^13) = (7^10 / 7^9) * (4^13 / 4^15)
Step 5: Simplifying the expression.
Using the exponent rule a^m / a^n = a^(m-n) again, we can simplify further:
(7^10 / 7^9) * (4^13 / 4^15) = 7^(10-9) * 4^(13-15) = 7^1 * 4^(-2) = 7/4^2 = 7/16
Therefore, the simplified fraction for the expression (4^3 7^-2)^-5 over 7^9 / 4^-13 with only positive exponents is 7/16.
To generate equivalent expressions for the given expression, we can apply the properties of integer exponents:
1. To start, let's simplify the expression (4^3 7^-2)^-5. First, we can evaluate the exponents within the parentheses:
4^3 = (4 * 4 * 4) = 64
7^-2 = 1/(7^2) = 1/49
So, the expression becomes (64 * (1/49))^-5.
2. Now, let's simplify the expression 7^9 / 4^-13. To do this, let's evaluate the exponents:
7^9 = (7 * 7 * 7 * 7 * 7 * 7 * 7 * 7 * 7) = 40353607
4^-13 = 1/(4^13)
So, the expression becomes (40353607) / (1/(4^13)).
3. Now, let's apply the rule that says any number raised to a negative exponent is equal to the reciprocal of the number raised to the positive exponent:
(40353607) / (1/(4^13)) = (40353607) * (4^13)
We can rewrite this expression as (40353607 * 4^13).
4. To simplify the expression further, we can use the property of multiplying exponents with the same base. In this case, the base is 4:
4^13 = (4^3)^4 = 64^4 = 16777216
So, the expression becomes (40353607 * 16777216).
5. To solve the expression, we simply multiply the numbers:
(40353607 * 16777216) = 677010714060912
6. Finally, we need to convert the result into a simplified fraction with no remaining exponents. Since the given expression does not involve any denominators, we can write the answer as a simplified fraction by setting the denominator to 1:
Answer: 677010714060912/1, which is the simplified fraction with no remaining exponents.