parenthesis 4 to the 3 power times 7 to the negative 2 power closed parenthesis negative 5 power / 7 to the 9th power times 4 to the negaitve 13 power Apply the Properties of Integer Exponents to generate equivalent expressions with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 poi
To apply the properties of integer exponents and generate equivalent expressions with only positive exponents, we can use the following rules:
1) When raising a product to an exponent, we can distribute the exponent to each factor:
(a * b)^n = a^n * b^n
2) When raising a power to another power, we can multiply the exponents:
(a^n)^m = a^(n * m)
Now let's simplify the expression step by step:
First, let's apply the rules mentioned above to simplify the expression inside the parentheses:
4^3 * 7^(-2)
Using the rule 1, we can distribute the exponents:
(4^3) * (7^(-2)) = 4^(3 * (-2)) = 4^(-6)
Now, we can rewrite the expression using positive exponents:
(7^9) * (4^(-13)) = (7^9) / (4^13)
Finally, we can simplify the expression by expanding the powers:
(7^9) / (4^13) = (7 * 7 * 7 * 7 * 7 * 7 * 7 * 7 * 7) / (4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4)
Now we can calculate the value of this expression. Since both the numerator and denominator have the same factors (7 and 4), they can cancel out:
(7 * 7 * 7 * 7 * 7 * 7 * 7 * 7 * 7) / (4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4)
= 7^9 / 4^13
= (7^9 / 4^9) / (4^4)
= (7/4)^9 / 256
Therefore, the simplified fraction with no remaining exponents is:
(7/4)^9 / 256
thats wrong
Apologies for the incorrect response. Let me correct the solution.
The given expression is:
(4^3 * 7^(-2)) / (7^9 * 4^(-13))
To simplify this expression, we can use the properties of exponents:
1) The negative exponent rule: If a number or variable is raised to a negative exponent, it can be rewritten as the reciprocal of the number or variable raised to the positive exponent.
2) The division of exponents rule: When dividing two numbers or variables with the same base raised to different exponents, we can subtract the exponents.
Using these rules, we can simplify the expression as follows:
(4^3 * 7^(-2)) / (7^9 * 4^(-13))
= (4^3 / 4^(-13)) * (7^(-2) / 7^9)
= 4^(3 - (-13)) * 7^(-2 - 9)
= 4^16 * 7^(-11)
Now we have a positive exponent. We can rewrite 7^(-11) as the reciprocal with a positive exponent:
4^16 / 1 * (1 / 7^11)
= 4^16 / 7^11
To simplify this fraction further, we need to see if any factors can cancel out. Both 4^16 and 7^11 are prime factors, so no cancellation is possible.
Therefore, our simplified expression is:
4^16 / 7^11
nope this is wrong too
Apologies for the incorrect response once again. Let's correct the solution.
The given expression is:
(4^3 * 7^(-2)) / (7^9 * 4^(-13))
To simplify this expression, we can apply the properties of exponents:
1) The power of a product rule: (ab)^n = a^n * b^n
2) The division of exponents rule: a^n / a^m = a^(n-m)
Using these rules, we can simplify the expression as follows:
(4^3 * 7^(-2)) / (7^9 * 4^(-13))
= (4^3 * 7^(-2)) / (4^(-13) * 7^9)
Now, let's simplify each numerator and denominator separately:
Numerator: (4^3 * 7^(-2))
= 4^(3) * 7^(-2)
= 4 * 4 * 4 * 1/(7 * 7)
Denominator: (4^(-13) * 7^9)
= 1/(4^13) * 7^9
= 1/(4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4) * (7 * 7 * 7 * 7 * 7 * 7 * 7 * 7 * 7)
Now, let's simplify the entire expression by canceling out common factors:
(4 * 4 * 4 * 1/(7 * 7)) / (1/(4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4) * (7 * 7 * 7 * 7 * 7 * 7 * 7 * 7 * 7))
= (4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4) * (7 * 7 * 7 * 7 * 7 * 7 * 7 * 7 * 7) / (7 * 7)
Cancelling out 7's and 4's, we have:
= 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 7 * 7 * 7 * 7 * 7 * 7 * 7 * 7
Now, we can find the product of all the remaining factors:
= 4^12 * 7^8
Therefore, the simplified expression is:
4^12 * 7^8
nope
Apologies for the errors in the previous responses. Let's correct the solution.
The given expression is:
(4^3 * 7^(-2)) / (7^9 * 4^(-13))
To simplify this expression, we can use the properties of exponents:
1) The power of a product rule: (ab)^n = a^n * b^n
2) The division of exponents rule: a^n / a^m = a^(n-m)
3) The negative exponent rule: a^(-n) = 1 / a^n
Using these rules, we can simplify the expression as follows:
(4^3 * 7^(-2)) / (7^9 * 4^(-13))
Let's first simplify the numerator:
4^3 * 7^(-2)
= 64 * (1/49)
= 64/49
Now let's simplify the denominator:
7^9 * 4^(-13)
= 7^9 * (1/4^13)
= 7^9 / 4^13
Now, we can combine the simplified numerator and denominator:
(64/49) / (7^9 / 4^13)
Using the division of exponents rule, we can rewrite this as:
(64/49) * (4^13 / 7^9)
Now, let's simplify the expression further:
= (2^6 / 7^2) * (2^26 / 7^9)
Using the division of exponents rule, we can rewrite this again as:
= (2^6 * 2^26) / (7^2 * 7^9)
= 2^(6+26) / 7^(2+9)
= 2^32 / 7^11
Therefore, the simplified expression is:
2^32 / 7^11
thats wrong too
Apologies for the previous incorrect responses. Let's correct the solution:
The given expression is:
(4^3 * 7^(-2)) / (7^9 * 4^(-13))
To simplify this expression, we can use the properties of exponents:
1) The power of a product rule: (ab)^n = a^n * b^n
2) The division of exponents rule: a^n / a^m = a^(n-m)
3) The negative exponent rule: a^(-n) = 1 / a^n
Using these rules, we can simplify the expression as follows:
First, let's simplify the numerator:
4^3 * 7^(-2)
= 64 * (1/49)
= 64/49
Now, let's simplify the denominator:
7^9 * 4^(-13)
= 7^9 / 4^13
Now, let's combine the simplified numerator and denominator:
(64/49) / (7^9 / 4^13)
Using the division of exponents rule and the negative exponent rule, we can rewrite this as:
(64/49) * (4^13 / 7^9)
= (64/49) * (4^13 * 1 / 7^9 * 1)
= (64/49) * (4^13 / 7^9)
= (64/49) * (2^26 / 7^9 * 2^26)
Using the power of a product rule:
= (64/49) * (2^26 * 2^26) / (7^9)
= (64/49) * (2^(26+26)) / (7^9)
= (64/49) * (2^52) / (7^9)
Now, let's simplify this expression further. We can leave the answer as a fraction with positive exponents:
= (64 * 2^52) / (49 * 7^9)
= (2^6 * 2^52) / (7^2 * 7^9)
= 2^(6+52) / 7^(2+9)
= 2^58 / 7^11
Therefore, the simplified expression is:
2^58 / 7^11