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.
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
5 answers
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).
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.
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.