Apply the Properties of Integer Exponents to generate equivalent expressions to (4^3 x 7^-2)^-5/7^9x4^-13 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
To generate equivalent expressions with only positive exponents, we can use the properties of integer exponents:
1. The Product Rule: (a^m * b^n)^p = a^(m * p) * b^(n * p)
2. The Quotient Rule: (a^m / b^n)^p = a^(m * p) / b^(n * p)
3. The Power Rule: (a^m)^n = a^(m * n)
4. Negative Exponent Rule: a^(-n) = 1 / a^n
Using these rules, we can simplify the given expression step by step:
(4^3 * 7^-2)^(-5) / 7^9 * 4^-13
Step 1: Simplify the exponents in the numerator:
4^3 = 64
7^-2 = 1 / 7^2 = 1 / 49
The numerator becomes: (64 * 1 / 49)^(-5)
Step 2: Simplify the exponents in the denominator:
7^9 = 7^9
4^-13 = 1 / 4^13
Therefore, the denominator remains the same: 7^9 * 1 / 4^13
Step 3: Apply the Power Rule to the numerator:
(64 * 1 / 49)^(-5) = (64 / 49)^(-5)
Step 4: Apply the Negative Exponent Rule to the numerator:
(64 / 49)^(-5) = (49 / 64)^5 = 49^5 / 64^5
Therefore, the expression becomes:
(49^5 / 64^5) / (7^9 * 1 / 4^13)
Step 5: Apply the Quotient Rule to divide the fractions:
(49^5 / 64^5) / (7^9 * 4^-13) = (49^5 / 64^5) * (4^13 / 7^9)
Step 6: Apply the Product Rule to simplify the expression:
(49^5 / 64^5) * (4^13 / 7^9) = (49^5 * 4^13) / (64^5 * 7^9)
Finally, we can solve the expression by evaluating the numerator and the denominator without exponents.
what is the answer
I apologize for the confusion. Let's solve the expression:
The numerator is (49^5 * 4^13).
To evaluate this, we have:
49^5 = 49 * 49 * 49 * 49 * 49 = 5764801
and
4^13 = 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 = 134217728.
Therefore, the numerator is 5764801 * 134217728 = 774840978915584.
The denominator is (64^5 * 7^9).
To evaluate this, we have:
64^5 = 64 * 64 * 64 * 64 * 64 = 1073741824
and
7^9 = 7 * 7 * 7 * 7 * 7 * 7 * 7 * 7 * 7 = 40353607.
Therefore, the denominator is 1073741824 * 40353607 = 43242803114664968.
Finally, the simplified expression is:
(49^5 * 4^13) / (64^5 * 7^9) = 774840978915584 / 43242803114664968
This cannot be further simplified as a fraction, so the final answer is:
774840978915584 / 43242803114664968