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.
To apply the properties of integer exponents, we can use the following rules:
1. (a^m)^n = a^(m*n)
2. a^(-n) = 1/a^n
3. a^0 = 1
4. a^1 = a
5. a^m/a^n = a^(m-n)
First, let's simplify the expression using the properties of integer exponents:
(4^3 ⋅ 7^(-2))^(-5)/(7^9 ⋅ 4^(-13))
Using rule 1, we can simplify the expression inside the parentheses:
4^(3*(-5)) ⋅ 7^((-2)*(-5))/(7^9 ⋅ 4^(-13))
This becomes:
4^(-15) ⋅ 7^10/(7^9 ⋅ 4^(-13))
Using rule 2, we can convert the negative exponents to positive exponents:
1/4^15 ⋅ 7^10/(7^9 ⋅ 4^(-13))
Using rule 5, we can simplify the expression further:
1/(4^15 ⋅ 4^(-13) ⋅ 7^(-9) ⋅ 7^10)
Using rule 1, we can combine the exponents with the same base:
1/(4^(15 + (-13)) ⋅ 7^(-9 + 10))
Simplifying the exponent calculations:
1/(4^2 ⋅ 7^1)
Using rule 4 and simplifying further:
1/(16 ⋅ 7)
Simplifying the expression:
1/112
Therefore, the simplified fraction is 1/112.
that is inncorrect 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.
To apply the properties of integer exponents, we can use the following rules:
1. Product Rule: (a^m ⋅ b^n)^p = a^(m⋅p) ⋅ b^(n⋅p)
2. Negative Exponent Rule: a^(-m) = 1/a^m
3. Quotient Rule: (a/b)^m = a^m / b^m
4. Power of a Power Rule: (a^m)^n = a^(m⋅n)
Let's simplify the expression step-by-step:
(4^3 ⋅ 7^(-2))^(-5/7^9) ⋅ 4^(-13)
First, apply the product rule inside the parentheses:
4^(3⋅(-5/7^9)) ⋅ 7^((-2)⋅(-5/7^9)) ⋅ 4^(-13)
Simplify the exponents:
4^(-15/7^9) ⋅ 7^(10/7^9) ⋅ 4^(-13)
Now, apply the quotient rule to the second term:
4^(-15/7^9) / 7^(-10/7^9) ⋅ 4^(-13)
Since we want to have positive exponents, we can use the negative exponent rule to move the terms to the denominator:
1 / (4^(15/7^9) ⋅ 7^(10/7^9)) ⋅ 4^(-13)
Now, simplify the exponents:
1 / (4^(15/7^9) ⋅ 7^(10/7^9)) ⋅ 4^(-13)
To combine the expressions in the denominator, we add the exponents:
1 / (4^(15/7^9 + (-13)) ⋅ 7^(10/7^9))
Simplify further:
1 / (4^((15 - 13⋅7^9)/7^9) ⋅ 7^(10/7^9))
Now, let's simplify the fraction in the exponent of 4:
1 / (4^(15 - 13⋅7^9)/7^9) ⋅ 7^(10/7^9)
To combine the expressions in the exponent of 7, we subtract the exponents:
1 / (4^(15 - 13⋅7^9)/7^9 ⋅ 7^(10 - 7^9)/7^9)
Simplify the exponents further:
1 / (4^(15 - 13⋅7^9)/7^9 ⋅ 7^(10 - 7^9)/7^9)
The expression is now in a simplified form.
To generate equivalent expressions with only positive exponents, we need to apply the properties of integer exponents:
1. Power of a Power: (a^m)^n = a^(mn)
2. Product of Powers: (ab)^n = a^n * b^n
3. Quotient of Powers: (a/b)^n = a^n / b^n
4. Power of a Product: (ab)^n = a^n * b^n
Let's simplify the given expression step by step:
(4^3 ⋅ 7^−2)^−5 / 7^9 ⋅ 4^−13
First, let's simplify the exponents within the parentheses using the power of a product property:
= 4^(3 * -5) ⋅ 7^(-2 * -5) / 7^9 ⋅ 4^(-13)
= 4^(-15) ⋅ 7^10 / 7^9 ⋅ 4^(-13)
Now, let's simplify the expression using the quotient of powers property:
= 4^(-15 + 13) ⋅ 7^(10 - 9)
= 4^(-2) ⋅ 7^1
Next, let's simplify further using the power of a power property:
= (4^(-1))^2 ⋅ 7^1
= (1/4)^2 ⋅ 7^1
= 1/4^2 ⋅ 7^1
Finally, let's simplify the expression:
= 1/16 ⋅ 7
= 7/16
Therefore, the simplified fraction with no remaining exponents is 7/16.