10. 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
*
1 point
7/16
21/28
-7/16
16
To apply the properties of integer exponents, we can use the rule that states for any nonzero number a, and integers m and n:
(a^m)^n = a^(m*n)
Using this rule, we can rewrite the expression (4^3 * 7^-2)^-5 as 4^(3*(-5)) * 7^(-2*(-5)). This becomes:
4^(-15) * 7^10
Now, to generate equivalent expressions with only positive exponents, we can use the rule that states for any nonzero number a:
a^-n = 1/a^n
Using this rule, we can rewrite 4^(-15) as 1/(4^15), and 7^10 as 1/(7^-10). This gives us:
1/(4^15) * 1/(7^-10)
Applying the rule that states for any nonzero numbers a and b:
1/a * 1/b = 1/(a*b)
We can simplify the expression further to:
1/(4^15 * 7^-10)
Now, to multiply the powers with the same base, we can use the rule that states for any nonzero number a, and integers m and n:
a^m * a^n = a^(m+n)
Using this rule, we can simplify the expression to:
1/(4^15 * 7^-10) = 1/(4^(15+10)*7^(-10))
So the final expression is:
1/(4^25 * 7^-10)
This can be written as a fraction with positive exponents by applying the rule that states for any nonzero number a:
a^-n = 1/a^n
Thus, 7^-10 becomes 1/7^10:
1/(4^25 * 1/7^10)
Multiplying the fractions:
1/(4^25 * 1/7^10) = 7^10/4^25
Simplifying further:
7^10/4^25 = (7/4)^10
So the equivalent expression with only positive exponents is (7/4)^10.
To solve this expression, we can raise 7/4 to the power of 10:
(7/4)^10 = 282475249/1048576
Thus, the simplified fraction with no remaining exponents is 282475249/1048576.
So the answer is: 282475249/1048576 (Option: D)
To solve the expression (4^3 * 7^-2)^-5, we can apply the properties of integer exponents in order to simplify the expression and only have positive exponents.
First, let's evaluate the exponents inside the parentheses:
4^3 = 4 * 4 * 4 = 64
7^-2 = 1/7^2 = 1/49
Substituting these values into the expression, we have:
(64 * 1/49)^-5
Now, let's simplify the expression inside the parentheses by multiplying the numbers:
64 * 1/49 = 64/49
Next, we can simplify the expression to have positive exponents by using the rule (a/b)^-n = (b/a)^n:
(64/49)^-5 = (49/64)^5
Now, we can raise the fraction (49/64) to the power of 5:
(49/64)^5 = 49^5 / 64^5
Finally, we can simplify the expression by expressing both numerator and denominator with prime factors:
49^5 = (7^2)^5 = 7^10
64^5 = (2^6)^5 = 2^30
Substituting these values back into the expression:
(49^5 / 64^5) = 7^10 / 2^30
Therefore, the simplified fraction with no remaining exponents is 7^10/2^30.
To simplify the expression (4^3 * 7^-2)^-5, we can start by applying the property of exponents that states when you raise a power to a negative exponent, you can switch the base and rewrite it with a positive exponent. This can be represented as follows:
(4^3 * 7^-2)^-5 = (4^3 / 7^2)^5
Next, we can simplify the expression inside the parentheses by evaluating the exponents:
= (64 / 49)^5
To raise a fraction to a power, we raise both the numerator and the denominator to the power:
= 64^5 / 49^5
Now we simplify the numerator and denominator by evaluating the exponents:
= 1073741824 / 8303765625
The final answer is the fraction 1073741824/8303765625.
Just the naswer pllz
The answer is 1073741824/8303765625.
That is not one of the choices
7/16
21/28
-7/16
16