Apply the Properties of Integer Exponents to generate equivalent expressions to (43 ⋅ 7−2)−579 ⋅ 4−13 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
To convert the expression to positive exponents using the properties of integer exponents, we can apply the following rules:
1) For any non-zero number a, a^(-n) is equal to 1/a^n.
2) For any non-zero number a, a^0 is equal to 1.
3) For any non-zero number a, a^m ⋅ a^n is equal to a^(m+n).
4) For any non-zero number a, (a^m)^n is equal to a^(m*n).
Applying these rules, we have:
(43 ⋅ 7^(-2))^(-579) ⋅ 4^(-13)
= (43 ⋅ 1/7^2)^(-579) ⋅ 1/4^13
= (43/49)^(-579) ⋅ 1/32768
Now, let's solve the expression:
= (1/(43/49)^579) ⋅ 1/32768
= 1/(1/(43/49)^579 ⋅ 32768)
= (43/49)^579 ⋅ 32768/1
The final result is (43/49)^579 ⋅ 32768.
To generate equivalent expressions with only positive exponents, we can use the properties of integer exponents:
1. Product of Powers: am ⋅ an = am+n
2. Quotient of Powers: am / an = am-n
3. Power of a Power: (am)n = amn
Let's apply these properties step-by-step to the expression (43 ⋅ 7^(-2))^(-579) ⋅ 4^(-13):
Step 1: Apply the power of a power property
(43 ⋅ 7^(-2))^(-579) ⋅ 4^(-13)
= (43)^(-579) ⋅ (7^(-2))^(-579) ⋅ 4^(-13)
Step 2: Apply the product of powers property to (43)^(-579)
= 43^(-579) ⋅ (7^(-2))^(-579) ⋅ 4^(-13)
Step 3: Apply the power of a power property to (7^(-2))^(-579)
= 43^(-579) ⋅ 7^((-2) * (-579)) ⋅ 4^(-13)
Step 4: Simplify the exponent: (-2) * (-579) = 1158
= 43^(-579) ⋅ 7^1158 ⋅ 4^(-13)
Now we have the expression with only positive exponents. Let's proceed to solve it:
Step 5: Evaluate the exponent of 43^(-579)
Since any non-zero number raised to the power of -n can be written as 1 / (that number)^n, we have:
43^(-579) = 1 / 43^579
Step 6: Simplify the expression further using the quotient of powers property
1 / 43^579 ⋅ 7^1158 ⋅ 4^(-13)
= (7^1158) / (43^579) ⋅ (1 / 4^13)
Step 7: Evaluate the exponent of 4^13
Since 4^(-n) can be written as 1 / (4^n), we have:
(1 / 4^13) = 1 / (4^13)
Putting it all together, the simplified form of the expression is:
(7^1158) / (43^579) ⋅ (1 / 4^13)
Please note that this is the simplified fraction form of the expression, and it cannot be reduced further without specific values for the variables involved.
To generate equivalent expressions with positive exponents, we need to apply the properties of integer exponents. The properties we will use are:
1. Product of Powers Property: When multiplying powers with the same base, we add the exponents.
2. Quotient of Powers Property: When dividing powers with the same base, we subtract the exponents.
3. Power of a Power Property: When raising a power to another power, we multiply the exponents.
4. Negative Exponent Property: A negative exponent indicates the reciprocal of the base raised to the positive exponent.
Let's apply these properties step by step to the given expression, (43 ⋅ 7^(-2))^(-579) ⋅ 4^(-13):
Step 1: Apply the Negative Exponent Property
(43 ⋅ 7^(-2))^(-579) ⋅ 4^(-13) can be rewritten as:
(43 ⋅ (1/7^2))^(-579) ⋅ (1/4^13)
Step 2: Simplify the exponents
(43 ⋅ (1/49))^(-579) ⋅ (1/4^13)
Step 3: Apply the Product of Powers Property
(43/49)^(-579) ⋅ (1/4^13)
Step 4: Apply the Power of a Power Property
[(43/49)^(-1)]^579 ⋅ (1/4^13)
Step 5: Take the reciprocal and simplify
[49/43]^579 ⋅ (1/4^13)
Now, let's solve the expression to get a simplified fraction with no remaining exponents.
(49/43)^579 ⋅ (1/4^13) ≈ 1.040570461 × 10^(-236)
The result is approximately 1.040570461 times 10 raised to the power of -236.