apply the properties of integer exponent to generate the quicalent expressions to: (4^3 x 7^-2) ^-5 /7^9 x 4 -13 with only positive exponents then solve the expression your answer will be simplified fraction with no remaining exponents
To apply the properties of integer exponents, let's simplify the given expression step by step.
(4^3 x 7^-2)^-5 / 7^9 x 4 - 13
First, let's focus on the base of 4^3 and 7^-2. We can use the property (a^m)^n = a^(m*n) to combine the exponents:
(4^3 x 7^-2)^-5 = 4^(3*-5) x 7^(-2*-5)
= 4^(-15) x 7^(10)
Now, let's simplify the expression further:
4^(-15) x 7^(10) / 7^9 x 4 - 13
Next, we can apply the division property a^m / a^n = a^(m-n) to simplify the expression:
4^(-15) x 7^(10) / 7^9 x 4 - 13 = 4^(-15) x 7^(10-9) x 4 - 13
= 4^(-15) x 7^1 x 4 - 13
Additionally, we notice that 4^-15 can be rewritten as 1/4^15:
(1/4^15) x 7^1 x 4 - 13
Now, let's apply the property (a^m x b^n) = (a x b)^(m+n) to combine the bases of 7^1 and 4:
(1/4^15) x 7^1 x 4 - 13 = (1/4^15 x 7 x 4) - 13
= (7/4^15 x 4) - 13
Finally, let's simplify the expression:
(7/4^15 x 4) - 13 = 7/4^15 - 13
Therefore, the simplified expression is 7/4^15 - 13.
nope please try a different way?
Apologies for the confusion. Let's try another approach to simplify the expression.
Given expression: (4^3 x 7^-2)^-5 / 7^9 x 4 - 13
First, let's simplify the base of (4^3 x 7^-2)^-5 using the property (a^m)^n = a^(m*n):
(4^3 x 7^-2)^-5 = 4^(3*-5) x 7^(-2*-5)
= 4^(-15) x 7^(10)
Now, let's rewrite 7^(10) using positive exponents by using the property a^-n = 1/a^n:
4^(-15) x 1/7^10
Next, let's simplify the expression by applying the division property a^m / a^n = a^(m-n) to divide the exponents:
4^(-15) x 1/7^10 = 4^(-15) / 7^10
Now, recall that a negative exponent in the denominator flips the base to the numerator and changes the sign of the exponent. Therefore, let's rewrite 4^(-15)/7^10 as 7^-10/4^15:
7^-10/4^15
Finally, let's simplify the expression using the property a^-n = 1/a^n:
1/7^10 x 1/4^15
To multiply these fractions, we multiply the numerators and denominators:
1 x 1 / 7^10 x 4^15
So, the simplified expression is 1/(7^10 x 4^15).
simplafy that more please
My apologies for not simplifying the expression further. Let's simplify it some more.
We had the expression 1/(7^10 x 4^15).
To simplify further, let's rewrite 7^10 and 4^15 in terms of their prime factorizations:
7^10 = (7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7)
4^15 = (2 x 2)^15 = (2^15 x 2^15) = 2^30
Now, let's substitute these values back into our expression:
1/(7^10 x 4^15) = 1/((7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7) x 2^30)
Since we are dividing by a product of the same base, we can rewrite it as a product of reciprocals:
1/(7^10 x 4^15) = 1/7^10 x 1/((7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7) x 2^30)
Now, we can cancel out common factors from the numerator and denominator:
1/(7^10 x 4^15) = 1/7^10 x 1/(7^10 x 2^30)
To simplify further, let's cancel out the common factors of 7^10 in the numerator and denominator:
1/(7^10 x 4^15) = 1/(7^10 x 2^30)
Finally, we have the simplified expression 1/(7^10 x 2^30).
wrong agian its (4^3 * 7^-2) ^5
--------------------
7^9 *4^-13
Apologies for the mistake. Let's simplify the given expression correctly:
(4^3 x 7^-2)^5 / 7^9 x 4^-13
First, let's focus on the base of (4^3 x 7^-2)^5. We can use the property (a^m)^n = a^(m*n) to combine the exponents:
(4^3 x 7^-2)^5 = 4^(3*5) x 7^(-2*5)
= 4^15 x 7^-10
Now, let's rewrite 4^-13 using the property a^-n = 1/a^n:
(4^15 x 7^-10) / 7^9 x 1/4^13
Next, let's multiply the numerators and denominators:
4^15 / 7^10 x 1/7^9 x 1/4^13
To multiply the bases with the same exponent, we can add their exponents:
4^15 / (7^10 x 7^9) x 1/4^13
Applying the property a^m x a^n = a^(m+n):
4^15 / 7^(10+9) x 1/4^13
Simplifying further:
4^15 / 7^19 x 1/4^13
Again, applying the property a^-n = 1/a^n:
1/4^13 x 4^15 / 7^19
Now, let's rewrite 4^15 using the property a^m/a^n = (a^m)/(a^n):
1/4^(13-15) x 4^15 / 7^19
Simplifying the exponents:
1/4^(-2) x 4^15 / 7^19
Applying a negative exponent to flip the base:
4^2 x 4^15 / 7^19
Simplifying further:
16 x 4^15 / 7^19
Therefore, the simplified expression is 16 x 4^15 / 7^19.