Apply the properties of integer exponents to generate equivalent expressions to (4^3 7^-2)^-5/7^9 x 4^-13 with only positive exponents then solve the expression, the answer will be a simplified fraction with no remaining exponents
First, let's apply the properties of integer exponents to generate equivalent expressions with positive exponents:
(4^3 7^-2)^-5/7^9 x 4^-13
= (4^3)^-5 (7^-2)^-5 7^-9 x 4^-13
= 4^-15 7^10 7^-9 x 4^-13
= 4^(-15-13) 7^(10-9)
= 4^-28 7^1
Now, let's simplify the expression.
The negative exponent of 4^-28 can be written as 1/4^28:
= 1/4^28 7
The expression is now in the form of a fraction. To combine the terms, we can rewrite 7 as 7/1:
= (1/4^28) (7/1)
= 7/4^28
Therefore, the simplified fraction with no remaining exponents is 7/4^28.
Incorrect
Apologies for the error. Let's try again:
Applying the properties of integer exponents:
(4^3 7^-2)^-5/7^9 x 4^-13
= 4^(3*(-5)) * 7^((-2)*(-5)) / 7^9 * 4^(-13)
= 4^-15 * 7^10 / 7^9 * 4^-13
Now, let's simplify the expression:
= (1/4^15) * (7^10 / 7^9) * (1/4^13)
= (1/4^15) * (7^(10-9)) * (1/4^13)
= (1/4^15) * (7^1) * (1/4^13)
= 7/4^15 * 1/4^13
= 7 / (4^15 * 4^13)
= 7 / 4^(15+13)
= 7 / 4^28
Therefore, the simplified fraction with no remaining exponents is 7 / 4^28.
To generate equivalent expressions with positive exponents, we can use the following properties of integer exponents:
1. Product of Powers: (a^m) * (a^n) = a^(m + n)
2. Quotient of Powers: (a^m) / (a^n) = a^(m - n)
3. Power of a Power: (a^m)^n = a^(m * n)
4. Negative Exponent: a^-n = 1 / a^n
Now, let's apply these properties step-by-step to the given expression:
(4^3 * 7^-2)^-5 / 7^9 * 4^-13
Step 1: Apply the Power of a Power property to (4^3 * 7^-2)^-5
[(4^3)^-5 * (7^-2)^-5] / 7^9 * 4^-13
Simplify further:
(4^-15 * 7^10) / 7^9 * 4^-13
Step 2: Apply the Product of Powers property to (4^-15 * 7^10) / 7^9
[(4^-15 * 7^10) * (7^9)^-1] * 4^-13
Simplify further:
(4^-15 * 7^10 * 7^-9) * 4^-13
Step 3: Combine like terms by applying the Product of Powers property to the bases with the same exponent:
(4^-15 * 4^-13) * (7^10 * 7^-9)
Simplify further:
4^-28 * 7
Step 4: Apply the Negative Exponent property to 4^-28 and rewrite as a positive exponent:
1 / 4^28 * 7
Step 5: Simplify the expression further, as no other simplifications are possible:
7 / 4^28
Therefore, the simplified expression is 7 / 4^28.
To generate equivalent expressions with only positive exponents, we can make use of the properties of integer exponents.
Let's break down the given expression step by step:
Expression: (4^3 7^-2)^-5/7^9 x 4^-13
1. Apply the exponent rule for a power raised to another power:
(4^3)^-5/7^9 x 7^4 x 4^-13
(4^(-5*3))/7^9 x 7^4 x 4^-13
2. Simplify the exponents:
(4^-15)/7^9 x 7^4 x 4^-13
3. Use the quotient rule for exponents:
4^(-15-13)/7^9 x 7^4
4. Simplify the exponents:
4^-28/7^9 x 7^4
5. Use the product rule for exponents:
(4^-28 x 7^4)/7^9
6. Simplify the exponents:
(1/4^28 x 7^4)/7^9
7. Apply the division of exponents:
1/(4^28 x 7^9) x 7^4
Now that we have the expression with only positive exponents, let's simplify it further:
To multiply two numbers with the same base, we add the exponents:
1/(4^28 x 7^9) x 7^4 = 1/(4^28 x 7^5)
Since we are asked to express the answer as a simplified fraction with no remaining exponents, we can rewrite 4^28 as (2^2)^28 and 7^5 as (7^2)^2 x 7.
1/(4^28 x 7^5) = 1/((2^2)^28 x (7^2)^2 x 7)
Using the property (a^b)^c = a^(b*c):
1/(4^28 x 7^5) = 1/(2^(56) x 7^(10) x 7)
Next, we can simplify by using the division of exponents:
1/(2^(56) x 7^(10) x 7) = 1/(2^(56)) x 1/(7^(10) x 7)
Therefore, the simplified expression is: 1/(2^56 x 7^11)
The answer to the expression, in a simplified fraction with no remaining exponents, is 1/(2^56 x 7^11).