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 and generate equivalent expressions with positive exponents, we can use the following rules:
1) For any non-zero number a, a^0 = 1.
2) For any non-zero number a, a^(-n) = 1/a^n.
3) For any non-zero number a, b, and integers m, n:
(a^m)^n = a^(m*n)
(a*b)^n = a^n * b^n
(a/b)^n = a^n / b^n
Let's simplify the expression using these properties:
(4^3 ⋅ 7^(-2))^(-5) / 7^9 ⋅ 4^(-13)
Step 1: Apply the properties inside the parentheses.
(4^3 ⋅ 7^(-2))^(-5) = (64 ⋅ 1/49)^(-5) = (64/49)^(-5)
Step 2: Simplify the exponents.
(64/49)^(-5) = (49/64)^5
Step 3: Apply the properties outside the parentheses.
(49/64)^5 / 7^9 ⋅ 4^(-13) = (49^5 / 64^5) / 7^9 ⋅ (1/4)^13
= 49^5 / 64^5 / 7^9 ⋅ 1/4^13
Step 4: Simplify the exponents and convert to positive exponents.
49^5 = 117,649
64^5 = 1,073,741,824
7^9 = 403,536,07
1/4^13 = 4^-13
Therefore, the expression simplifies to:
117,649 / 1,073,741,824 / 403,536,07 ⋅ 4^-13
Simplifying further, we get:
117,649 / (1,073,741,824 * 403,536,07) ⋅ 1/4^13
= 117,649 / (433,016,202,826,009,088) ⋅ 1/4^13
Now, we need to simplify this fraction with no remaining exponents.
117,649 / (433,016,202,826,009,088) ⋅ 1/4^13
= 117,649 / (433,016,202,826,009,088) ⋅ 1/(2^13 * 2^13)
= 117,649 / (433,016,202,826,009,088) ⋅ 1/(2^26)
To simplify the fraction, we can divide the numerator and denominator by 117,649.
= 1 / (3,685,904) ⋅ 1/(2^26)
So the simplified expression is:
1 / (3,685,904) ⋅ 1/(2^26)
This cannot be simplified further.
It can though. get off your computer as and help
Apologies for the confusion. Let's simplify the expression further.
To simplify (1 / (3,685,904)) ⋅ 1/(2^26), we can rewrite 1 / (3,685,904) as (3,685,904)^(-1) using the property that a^(-1) = 1/a.
So the expression becomes (3,685,904)^(-1) ⋅ 1/(2^26).
Now, we can simplify the exponents:
(3,685,904)^(-1) = 1 / (3,685,904)
1 / (2^26) = 1 / (2^(13*2)) = 1 / (2^13 * 2^13) = 1 / (2^26)
Therefore, the expression simplifies to:
1 / (3,685,904) ⋅ 1 / (2^26) = (1 / (3,685,904)) ⋅ (1 / (2^26)) = (3,685,904)^(-1) ⋅ (2^(-26))
Since both terms have the same base (2), we can combine them:
(3,685,904)^(-1) ⋅ (2^(-26)) = (3,685,904 * 2^(-26))^(-1)
So our final simplified expression is:
(3,685,904 * 2^(-26))^(-1), where 2^(-26) means 1 / (2^26).
think again u piece of cra-p
I apologize if my previous response did not meet your expectations. Let's try simplifying the expression again.
The expression is (4^3 ⋅ 7^(-2))^(-5) / 7^9 ⋅ 4^(-13).
First, let's simplify the exponents inside the parentheses:
4^3 = 64
7^(-2) = 1/7^2 = 1/49
Thus, the expression becomes (64 ⋅ 1/49)^(-5) / 7^9 ⋅ 4^(-13).
Next, simplify the expression further:
(64 ⋅ 1/49)^(-5) = (64/49)^(-5)
Now, let's simplify the exponents:
(64/49)^(-5) = (49/64)^5
Continuing to simplify, we can rewrite the expression as:
(49^5 / 64^5) / 7^9 ⋅ 4^(-13) = (49^5 / 64^5) / 7^9 / 4^13
To further simplify, compute the values of the numbers involved:
49^5 = 49 * 49 * 49 * 49 * 49 = 576,480,1
64^5 = 64 * 64 * 64 * 64 * 64 = 1,073,741,824
Hence, the expression simplifies to:
576,480,1 / 1,073,741,824 / 7^9 / 4^13
Now, applying the properties of exponents, we can rewrite the expression with positive exponents:
576,480,1 / 1,073,741,824 / 7^9 / 4^13 = 576,480,1 / 1,073,741,824 / 7^9 / (2^2)^13 = 576,480,1 / 1,073,741,824 / 7^9 / 2^26
To simplify this expression further, let's divide the numerator and denominator by 576,480,1:
1 / 1,073,741,824 / 7^9 / 2^26 = 1 / 1,073,741,824 * 576,480,1 / 7^9 / 2^26 = 576,480,1 / (1,073,741,824 * 7^9 * 2^26)
Finally, we have simplified the expression to:
576,480,1 / (1,073,741,824 * 7^9 * 2^26)
This cannot be further simplified as there are no common factors between the numerator and denominator.
To generate equivalent expressions with only positive exponents, we can apply the properties of integer exponents:
1. (a^m)^n = a^(m * n) - The power of a power property.
2. a^m / a^n = a^(m - n) - The quotient of powers property.
Let's simplify the expression step-by-step:
(4^3 ⋅ 7^(-2))^(-5) / 7^9 ⋅ 4^(-13)
Step 1: Simplify the exponents within the parentheses:
4^(3 * -5) ⋅ 7^(-2 * -5) / 7^9 ⋅ 4^(-13)
Step 2: Simplify the exponents, keeping in mind that a^(-n) = 1 / a^n:
4^(-15) ⋅ 7^10 / 7^9 ⋅ 4^(-13)
Step 3: Apply the quotient of powers property:
4^(-15 - (-13)) ⋅ 7^(10 - 9)
Simplifying the exponents further:
4^(-15 + 13) ⋅ 7^1
Step 4: Simplify the exponents and perform calculations:
4^(-2) ⋅ 7
Step 5: Apply the definition of negative exponents:
1 / 4^2 ⋅ 7
Step 6: Perform the calculations:
1 / 16 ⋅ 7
Step 7: Simplify the fraction:
7 / 16
Therefore, the simplified fraction with no remaining exponents is 7/16.
To generate equivalent expressions with only positive exponents, we can apply the properties of integer exponents. The properties we are going to use are as follows:
1. Product Rule: a^m ⋅ a^n = a^(m+n)
2. Quotient Rule: a^m / a^n = a^(m-n)
3. Power Rule: (a^m)^n = a^(m⋅n)
4. Negative Exponent Rule: a^(-n) = 1 / a^n
Now let's apply these properties step by step:
Step 1: Rewrite the given expression using the properties of exponents.
(4^3 ⋅ 7^(-2))^(-5) / 7^9 ⋅ 4^(-13)
Using properties, we get:
(4^3 ⋅ 7^(-2))^(-5) = 4^(-5 ⋅ 3) ⋅ 7^(2 ⋅ 5) = 4^(-15) ⋅ 7^10
7^9 ⋅ 4^(-13) = 7^9 ⋅ (1/4^13) = 7^9/(4^13)
Step 2: Combine the generated expressions with positive exponents.
(4^(-15) ⋅ 7^10) / (7^9/(4^13))
Step 3: Simplify the expression further.
Recall that a^m / a^n = a^(m-n)
Using the quotient rule, we have: (4^(-15) ⋅ 7^10) / (7^9/(4^13)) = 4^(-15) ⋅ 7^10 ⋅ 4^13 / 7^9
Now, apply the product rule: a^m ⋅ a^n = a^(m+n)
So, 4^(-15) ⋅ 7^10 ⋅ 4^13 / 7^9 = 4^(-15 + 13) ⋅ 7^(10 - 9) = 4^(-2) ⋅ 7^1 = 1/4^2 ⋅ 7
Simplifying further, we get:
1/4^2 ⋅ 7 = 1/16 ⋅ 7 = 7/16
Therefore, the simplified fraction without remaining exponents is 7/16.