Apply the Properties of Integer Exponents to generate equivalent expressions to (7^−3)^7 ⋅ 7^20/6^−2 with only positive exponents, then solve the expression. Provide the answer as an improper fraction.
To generate equivalent expressions with only positive exponents, we can use the following properties of integer exponents:
1. Negative exponent property: For any nonzero number a, a⁻ⁿ = 1/aⁿ.
2. Product of powers property: (aⁿ)ᵐ = aⁿᵐ.
3. Quotient of powers property: (a/b)ⁿ = aⁿ/bⁿ.
Let's apply these properties to the given expression:
(7^−3)^7 ⋅ 7^20/6^−2
First, simplify the exponent expression (7^−3)^7 using the negative exponent property:
(7^−3)^7 = (1/7³)^7 = (1/343)^7
Now, simplify the base expression 7^20 using the product of powers property:
7^20 = 7^(6+6+6+2) = 7^6 ⋅ 7^6 ⋅ 7^6 ⋅ 7² = (7^6)³ ⋅ 7²
Next, simplify 6^−2 using the negative exponent property:
6^−2 = 1/6² = 1/36
Now, substitute these simplified expressions back into the original expression:
(1/343)^7 ⋅ (7^6)³ ⋅ 7² / (1/36)
Using the quotient of powers property, we can rewrite (1/343)^7 as 1^7 / 343^7:
(1^7 / 343^7) ⋅ (7^6)³ ⋅ 7² / (1/36)
Simplifying, we get:
1/343^7 ⋅ 7^18 ⋅ 7² / (1/36)
Now, we can rewrite 1/343^7 as 343^-7:
343^-7 ⋅ 7^18 ⋅ 7² / (1/36)
Using the product of powers property, we can combine the exponents:
343^(-7+18) ⋅ 7² / (1/36)
Simplifying the exponents:
343^11 ⋅ 7² / (1/36)
Now, rewrite 343^11 as (7²)^11:
(7²)^11 ⋅ 7² / (1/36)
Using the product of powers property, we can multiply the exponents:
7^(2*11) ⋅ 7² / (1/36)
Simplifying the exponents:
7^22 ⋅ 7² / (1/36)
Using the product of powers property:
7^(22+2) / (1/36)
Simplifying:
7^24 / (1/36)
Finally, we can rewrite the expression using positive exponents and solve it:
36 ⋅ 7^24
Therefore, the equivalent expression with only positive exponents is 36 ⋅ 7^24, and it cannot be simplified further.
are you sure?
Apologies for the confusion in my previous response. Let's go through the simplification again:
Given expression: (7^−3)^7 ⋅ 7^20 / 6^−2
Apply the negative exponent property to (7^−3)^7:
(7^−3)^7 = 7^(−3 * 7) = 7^−21
Apply the negative exponent property to 6^−2:
6^−2 = 1 / 6^2 = 1 / 36
Now, substitute the simplifications back into the original expression:
7^−21 ⋅ 7^20 / 1 / 36
Using the product of powers property, combine the exponents:
7^(−21 + 20) / 1 / 36
Simplify the exponents:
7^−1 / 1 / 36 = 1 / 7 / 1 / 36 = 36 / 7
Therefore, the expression (7^−3)^7 ⋅ 7^20 / 6^−2 simplifies to the improper fraction 36 / 7.