pply the Properties of Integer Exponents to generate equivalent expressions to (4^3⋅7^−2) ^−5/7^9 ⋅ 4^−13 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point
To generate equivalent expressions with only positive exponents, we can use the properties of integer exponents.
First, let's simplify the expression:
(4^3⋅7^−2) ^−5/7^9 ⋅ 4^−13
Using the property of negative exponents, we can rewrite the expression as:
(4^3/7^2)^-5/7^9 ⋅ 4^(-13)
Now, let's simplify the exponents:
(64/49)^(-5/49) ⋅ (1/4^13)
Using the property of raising a fraction to a negative exponent, we can rewrite the expression as:
(49/64)^(5/49) ⋅ 1/(2^26)
Now, let's simplify the expression further by converting to a single fraction:
(49/64)^(5/49) / (2^26)
Now, let's simplify the exponents:
49^(5/49) / 64^(5/49) / 2^26
Now, let's simplify the base and exponent of 49^(5/49):
7^5 / 64^(5/49) / 2^26
Now, let's simplify the base and exponent of 64^(5/49):
7^5 / 4^(5/7) / 2^26
Finally, let's simplify the base and exponent of 4^(5/7):
7^5 / 2^(5/7) / 2^26
We can combine the exponents of the denominator:
7^5 / 2^(5/7 + 26)
Now, let's simplify the exponent of the denominator:
7^5 / 2^(5/7 + 182/7)
Now, let's simplify the exponent further:
7^5 / 2^(187/7)
Now, let's simplify the expression further by converting the denominator to a fraction with a common denominator:
7^5 / 2^(187/7)
Now, let's convert the denominator to a radical:
7^5 / √(2^187)
Finally, let's simplify the expression further by evaluating the square root and raising it to the 7th power:
7^5 / (2^26 √2)
Therefore, the simplified expression is 7^5 / (2^26 √2).
are you sure
Apologies for the mistake in my previous response. Here is the correct solution:
To generate equivalent expressions with only positive exponents, we can use the properties of integer exponents.
First, let's simplify the expression:
(4^3⋅7^−2) ^−5/7^9 ⋅ 4^−13
Using the property of negative exponents, we can rewrite the expression as:
(4^3/7^2) ^−5/7^9 ⋅ 1/(4^13)
Now, let's simplify the exponents:
(64/49)^(-5/49) ⋅ 1/(2^52)
Next, we can apply the property of negative exponents to rewrite (64/49)^(-5/49) as (49/64)^(5/49).
This gives us:
(49/64)^(5/49) ⋅ 1/(2^52)
Now, let's simplify further by expressing (49/64)^(5/49) as a radical:
√(49/64)^(5) ⋅ 1/(2^52)
Simplifying the radicals:
(7/8)^5 ⋅ 1/(2^52)
Now, we can simplify the expression by multiplying the exponents:
7^5 / 8^5 ⋅ 1/(2^52)
Finally, we can simplify further by evaluating the exponents:
16807 / 32768 ⋅ 1/(2^52)
Therefore, the simplified expression is 16807/(32768⋅2^52).
wrong
Apologies for the confusion. Let's try again:
To generate equivalent expressions with only positive exponents, we can use the properties of integer exponents.
First, let's simplify the expression:
(4^3⋅7^−2) ^−5/7^9 ⋅ 4^−13
Using the property of negative exponents, we can rewrite the expression as:
(4^3/7^2) ^−5/7^9 ⋅ 1/(4^13)
Now, let's simplify the exponents:
(64/49)^(-5/49) ⋅ 1/(2^52)
Using the property of raising a fraction to a negative exponent, we can rewrite the expression as:
(49/64)^(5/49) ⋅ 1/(2^52)
Now, let's simplify the expression further:
49^(5/49)/64^(5/49) ⋅ 1/(2^52)
Since the base and exponent of 49^(5/49) are the same, the exponent simplifies to 1:
49/64^(5/49) ⋅ 1/(2^52)
Since the exponent of 64^(5/49) is also 1, the expression becomes:
49/64 ⋅ 1/(2^52)
Now, let's simplify the expression further:
49/64 ⋅ 1/(2^52)
Now, let's simplify the denominator:
49/64 ⋅ 1/(2^52)
Now, let's simplify the exponent of 2:
49/64 ⋅ 1/(2^(52/1))
Now, let's simplify the exponent further:
49/64 ⋅ 1/(2^52)
Now, let's evaluate the expression:
49/64 ⋅ 1/(4,503,599,627,370,495,609,375)
To simplify the expression, we can multiply the numerator and denominator:
49 / (64 * 4,503,599,627,370,495,609,375)
Finally, let's write the simplified fraction:
49 / 288,230,376,151,711,744,052,585,840,123,873,750
Therefore, the simplified expression is 49 / 288,230,376,151,711,744,052,585,840,123,873,750.