Apply the properties of integer exponents to generate equivalent expressions to (4^3 x 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.
To apply the properties of integer exponents, we can use the following rules:
1) (a^m)^n = a^(m*n)
2) (a*b)^m = a^m * b^m
3) a^(-n) = 1/a^n
Let's simplify the given expression step by step:
(4^3 x 7^-2)^-5 / 7^9 x 4^13
Using the rule (a^m)^n = a^(m*n), we can simplify (4^3 x 7^-2)^-5 to (4^(3*-5) x 7^(-2*-5)).
Now the expression becomes:
(4^(3*-5) x 7^(-2*-5)) / 7^9 x 4^13
Using the rule a^(-n) = 1/a^n, we can simplify 7^(-2*-5) to 1/7^(2*5).
Now the expression becomes:
(4^(3*-5) x 1/7^(2*5)) / 7^9 x 4^13
Let's simplify the exponents:
(4^(-15) x 1/7^10) / 7^9 x 4^13
Now we can combine the division of exponents using the rule (a/b)^m = a^m / b^m:
4^(-15) / 7^10 x 7^-9 x 4^13
Using the rule (a*b)^m = a^m * b^m, we can rearrange the factors:
(4^(-15) x 4^13) / (7^10 x 7^9)
Now simplify each term:
4^(-15+13) / 7^(10+9)
4^(-2) / 7^19
Using the rule a^(-n) = 1/a^n, we can rewrite 4^(-2) as 1/4^2:
1/4^2 / 7^19
Simplifying further:
1/16 / 7^19
To divide fractions, we multiply the numerator by the reciprocal of the denominator:
1/16 * 1/7^19
Multiplying the numerators and denominators:
1/112^19
Now the expression is simplified and there are no remaining exponents.
To generate equivalent expressions with only positive exponents, we can start by using the 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)
Let's apply these properties step by step to simplify the expression:
Step 1: Simplify (4^3 x 7^-2)^-5
We can simplify the base separately using the product of powers property:
4^3 = 4^(3*1) = 4^3
7^-2 = 1/(7^2) = 1/49
Now, substitute the simplified forms back into the expression:
(4^3 x 7^-2)^-5 = (4^3 x 1/49)^-5
Step 2: Simplify 7^9 x 4^13
We can use the product of powers property:
7^9 x 4^13 = 7^9 x 4^(9*4) = 7^9 x 4^36
Now, we have simplified the given expression to:
(4^3 x 1/49)^-5 / (7^9 x 4^36)
Step 3: Combine the expression
Now, we can simplify further by applying the quotient of powers property:
(4^3 x 1/49)^-5 / (7^9 x 4^36) = (4^(3*(-5)) x (1/49)^(-5)) / (7^9 x 4^36)
Simplify the exponents:
(4^(-15) x 49^5) / (7^9 x 4^36)
Step 4: Simplify the expression
Let's simplify each base separately:
4^(-15) = 1/(4^15)
49^5 = 7^10
4^36 = (2^2)^36 = 2^(2*36) = 2^72
Substitute the simplified forms back into the expression:
(1/(4^15) x 7^10) / (7^9 x 2^72)
Simplify 7^10 / 7^9 = 7^(10-9) = 7^1 = 7
The final simplified expression is:
1/(4^15 x 7 x 2^72)
This cannot be simplified any further, so the answer is 1/(4^15 x 7 x 2^72).
To generate equivalent expressions with only positive exponents, we can apply the properties of integer exponents. The properties we'll use are:
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)
Let's simplify the expression step by step:
Step 1: Apply the product rule to (4^3)*(7^-2):
(4^3) * (7^-2) = 4^(3+(-2)) = 4^1 = 4
Step 2: Apply the power rule to (4)*(7^9):
(4)*(7^9) = 4*7^9
Step 3: Apply the quotient rule to (4*7^9)/(4^13):
(4*7^9)/(4^13) = 7^9 / 4^(13-1) = 7^9 / 4^12
Step 4: Combine the previous results:
(4^3 * 7^-2)^(-5) / 7^9 * 4^13 = (4)^(-5) * (7^9)/(4^12) * 4^13
= 4^(-5) * 7^9 * 4^13 / 4^12
= 4^(-5) * 7^9 * 4^(13-12)
= 4^(-5) * 7^9 * 4^1
= 4^(-5+1) * 7^9
= 4^(-4) * 7^9
Step 5: Simplify the expression:
4^(-4) * 7^9 = (1/4^4) * 7^9 = 1/256 * 7^9
Hence, the simplified fraction is 1/256 * 7^9.
thats wrong
I apologize for the error in my previous response. Let's go through the steps correctly this time:
Step 1: Apply the product rule to (4^3) * (7^-2):
(4^3) * (7^-2) = 4^(3+(-2)) = 4^1 = 4
Step 2: Apply the power rule to (4) * (7^9):
(4) * (7^9) = 4 * 7^9
Step 3: Apply the quotient rule to (4 * 7^9) / (4^13):
(4 * 7^9) / (4^13) = (7^9) / (4^(13-1)) = (7^9) / (4^12)
Step 4: Combine the previous results:
(4^3 * 7^-2)^(-5) / (7^9 * 4^13) = (4)^(-5) * (7^9)/(4^12)
= 1/(4^5) * 7^9/4^12
= 1/4^5 * 7^9/4^12
Now, let's simplify the exponents:
1/4^5 * 7^9/4^12 = 1/1024 * 7^9/4^12
Notice that 4^12 = (2^2)^12 = 2^24, so:
1/1024 * 7^9/4^12 = 1/1024 * 7^9/2^24
The expression is now in terms of positive exponents only, and cannot be further simplified. The simplified fraction is 1/1024 * 7^9/2^24.