Apply the properties of integer exponents to generate an expression equivalent to (4^3×7^-2)^-5/7^9×4^-13 with only positive exponents then solve the expression
It will be a simplified fraction with no remaining exponents
To apply the properties of integer exponents, we can multiply exponents when raising a power to a power, divide exponents when dividing powers with the same base, and subtract exponents when dividing powers with the same base but different exponents.
First, let's simplify the expression (4^3 × 7^(-2))^(-5) / (7^9 × 4^(-13)).
Applying the power of a product: (4^3 × 7^(-2))^(-5) is equal to 4^(3*(-5)) × 7^((-2)*(-5)).
So now our expression becomes:
4^(-15) × 7^10 / (7^9 × 4^(-13)).
Applying the power of quotient: 4^(-15) / 4^(-13) is equal to 4^((-15)-(-13)).
Now our expression becomes:
4^((-15)-(-13)) × 7^10 / 7^9.
Simplifying:
4^((-15)-(-13)) simplifies to 4^(-15+13) which is 4^(-2).
7^10 / 7^9 simplifies to 7^(10-9) which is 7^1.
Now our expression becomes:
4^(-2) × 7^1.
Finally, evaluating this expression:
4^(-2) is equal to 1/4^2 which is 1/16.
7^1 is equal to 7.
Putting it all together, the simplified expression is:
(1/16) × 7 which simplifies to 7/16.
9.(-33)0+3/5(8-27 0)
Apply the Zero Power Rule to create an equivalent expression to:
To apply the properties of integer exponents, let's simplify the expression step by step:
Step 1: Simplify the exponents within parentheses (4^3 × 7^-2)^-5:
Inside the parentheses:
4^3 = 4 × 4 × 4 = 64
7^-2 = 1 / (7^2) = 1 / (7 × 7) = 1 / 49
So, (4^3 × 7^-2)^-5 = (64 × 1/49)^-5
Step 2: Simplify the exponents outside parentheses:
The expression becomes (64 × 1/49)^-5 / 7^9 × 4^-13
64^-5 / 49^-5 = (1 / 64^5) / (1 / 49^5) = (49^5) / (64^5)
So, the expression simplifies to (49^5) / (64^5) / 7^9 × 4^-13
Step 3: Convert all exponents to positive:
(49^5) / (64^5) / 7^9 × 4^-13 = (49^5) / (64^5) / (7^9 × 4^13)
Step 4: Simplify the expression:
The expression is now (49^5) / (64^5) / (7^9 × 4^13)
To solve this expression, we must evaluate the values of 49^5, 64^5, 7^9, and 4^13.
Using a calculator:
49^5 ≈ 1,625,204,328,049
64^5 ≈ 1,073,741,824
7^9 ≈ 4,782,969
4^13 ≈ 67,108,864
So, the expression becomes:
(1,625,204,328,049) / (1,073,741,824) / (4,782,969 × 67,108,864)
Further simplification might be possible based on specific instructions, but at this stage, we have a simplified fraction with no remaining exponents.