Apply the Properties of Integer Exponents to generate equivalent expressions to (4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
To apply the properties of integer exponents, we can start by simplifying the expression inside the parentheses:
(4^3 ⋅ 7^−2)^−5 = 4^(3*-5) ⋅ 7^(-2*-5) = 4^(-15) ⋅ 7^10
Now, we can rewrite the expression using positive exponents:
(4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13 = (4^(-15) ⋅ 7^10)/(7^9 ⋅ 4^13)
Simplifying further:
= 4^(-15-13) ⋅ 7^(10-9)
= 4^(-28) ⋅ 7^1
Now, to solve the expression, we need to evaluate 4^(-28) and 7^1:
4^(-28) = 1/4^28 (since any number raised to the power of -n is equal to 1 divided by that number raised to the power of n)
7^1 = 7
So the expression becomes:
(1/4^28) ⋅ 7
= 7/4^28
To generate equivalent expressions with only positive exponents, we can apply the properties of integer exponents.
First, let's apply the property (a^m / b^n)^p = a^(m * p) / b^(n * p) to simplify the expression (4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13:
= [(4^3)^(-5/7) ⋅ (7^(-2))^(-5/7)] / (7^9)^(-5/7) ⋅ 4^(-13)
Next, let's simplify the exponents using the property (a^m)^n = a^(m * n):
= (4^(3 * (-5/7)) ⋅ (7^(-2 * (-5/7)))) / (7^(9 * (-5/7))) ⋅ 4^(-13)
Simplifying the exponents further:
= (4^(-15/7) ⋅ 7^(10/7)) / 7^(-45/7) ⋅ 4^(-13)
Now, let's apply the property a^(-n) = 1 / a^n to rewrite the expression with positive exponents:
= (7^(10/7) ⋅ 4^(13)) / (4^(15/7) ⋅ 7^(45/7))
Finally, let's simplify the expression:
= (7^(10/7 + 45/7) ⋅ 4^(13 - 15/7)) / 4^(15/7)
= (7^(55/7) ⋅ 4^(76/7)) / 4^(15/7)
= 7^(55/7) ⋅ 4^(76/7 - 15/7)
= 7^(55/7) ⋅ 4^(61/7)
Therefore, the equivalent expression with only positive exponents is 7^(55/7) ⋅ 4^(61/7).
To generate equivalent expressions with positive exponents, we can apply the properties of integer exponents:
1. Product Rule: (a^m * b^n)^p = a^(m*p) * b^(n*p)
2. Quotient Rule: (a^m / b^n)^p = a^(m*p) / b^(n*p)
3. Power Rule: (a^m)^p = a^(m*p)
Now let's apply these rules step by step to transform the given expression:
Step 1: Convert negative exponents to positive exponents
To convert 7^(-2) to a positive exponent, we use the reciprocal property: a^(-n) = 1 / a^n. Therefore, 7^(-2) can be written as 1 / 7^2.
Step 2: Simplify the exponents
Inside the parentheses, simplify the exponents first:
(4^3 ⋅ 1 / 7^2)^(-5/7^9 ⋅ 4^(-13))
Step 3: Apply the Power Rule
Apply the power rule to both bases:
4^(3 * -5/7^9) * (1 / 7^2)^(-5/7^9) * 4^(-13)
Step 4: Simplify the exponents further
4^(-15/7^9) * (1 / 7^(-10/7^9)) * 4^(-13)
Step 5: Apply the Quotient Rule
Divide the exponents:
4^(-15/7^9 - (-10/7^9)) * 4^(-13)
Step 6: Simplify the exponents
4^(-15/7^9 + 10/7^9) * 4^(-13)
4^(-5/7^9) * 4^(-13)
Step 7: Apply the Product Rule
Multiply the exponents:
4^((-5-13)/7^9)
4^(-18/7^9)
Finally, we have the transformed expression: 4^(-18/7^9)
To solve this expression, we need to simplify the exponent. To do this, we can convert the negative exponent into a positive exponent:
1 / 4^(18/7^9)
This is the simplified fraction form of the given expression, where the negative exponents have been eliminated.