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.
(4^3 * 7^-2)^-5
7^9 * 4^-13
7/16
21/28
-7/16
16
To apply the properties of integer exponents, we can use the rule that states (a^m)^n = a^(m*n). We can also use the rule that a^(-m) = 1/(a^m).
Starting with the expression (4^3 * 7^-2)^-5, we can first simplify the expression inside the parentheses using the rule stated above:
(4^3 * 7^-2)^-5 = 4^(3 * -5) * 7^(-2 * -5) = 4^(-15) * 7^10
Next, we can rewrite the expression with only positive exponents by using the second rule stated above:
4^(-15) * 7^10 = 1/(4^15) * 7^10 = 7^10 / 4^15
To simplify the fraction, we can rewrite 7^10 as (7^2)^5 and 4^15 as (2^2)^15:
7^10 / 4^15 = (7^2)^5 / (2^2)^15 = 49^5 / 2^30
Since both 49 and 2 are not perfect powers, we cannot further simplify the expression. Therefore, the final answer is 49^5 / 2^30, which is a simplified fraction with no remaining exponents.
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.
(4^3 * 7^-2)^-5 / 7^9 * 4^-13
7/16
21/28
-7/16
16
To solve the expression (4^3 * 7^-2)^-5 / 7^9 * 4^-13, we can apply the properties of integer exponents.
First, let's simplify the expression inside the parentheses:
(4^3 * 7^-2) = 4^3 * (1/7^2) = 4^3 / 7^2
Next, let's rewrite the expression without negative exponents:
(4^3 / 7^2)^-5 = (7^2 / 4^3)^5
Now, we can simplify the expression by applying the exponents to the numerator and denominator:
(7^2 / 4^3)^5 = 7^(2*5) / 4^(3*5) = 7^10 / 4^15
Next, let's simplify the expression outside the parentheses:
7^9 * 4^-13 = 7^9 / 4^13
Finally, let's divide the two expressions:
(7^10 / 4^15) / (7^9 / 4^13)
When dividing fractions, we can multiply the numerator by the reciprocal of the denominator:
(7^10 / 4^15) * (4^13 / 7^9)
We can now simplify the expression by canceling out common factors:
(7^10 * 4^13) / (4^15 * 7^9) = 7^(10-9) * 4^(13-15) = 7^1 * 4^-2 = 7/16
Therefore, the simplified fraction for the expression (4^3 * 7^-2)^-5 / 7^9 * 4^-13 is 7/16.
To generate an equivalent expression with only positive exponents, we'll use the properties of integer exponents.
First, let's simplify the expression inside the parentheses (4^3 * 7^-2):
Since 7^-2 is the reciprocal of 7^2, we can rewrite it as 1/7^2. Thus, the expression becomes:
4^3 * 1/7^2
Now, let's simplify this expression further:
4^3 is equal to 4 * 4 * 4, which is 64.
And 1/7^2 is equal to 1/49.
Therefore, the expression simplifies to:
64 * 1/49
Now, let's solve the expression (64 * 1/49)^-5:
Since a negative exponent indicates taking the reciprocal of the base raised to the positive exponent, we can rewrite the expression as:
(1/64 * 49)^5
Now, let's simplify this further:
1/64 * 49 is equal to 49/64.
And raising (49/64) to the power of 5 means multiplying it by itself five times:
(49/64)^5 = (49/64) * (49/64) * (49/64) * (49/64) * (49/64)
Now, let's calculate the result:
(49/64) * (49/64) * (49/64) * (49/64) * (49/64) = 5764801/11239424
Therefore, the final answer is the simplified fraction 5764801/11239424.