apply the properties of integer exponent to generate equivalent expressions to (7^3)^7 times 7^20 divided by 6^-2 with only positive exponents, then solve the expression. provide the answer as an improper fraction

To simplify the given expression using the properties of integer exponents, we'll follow these steps:

Step 1: Simplify the expression (7^3)^7.
According to the properties of exponents, when raising a power to another power, we multiply the exponents. Therefore, (7^3)^7 can be simplified as 7^(3*7) = 7^21.

Step 2: Simplify the expression 7^20.
Since the base is the same (7) and the exponents are being multiplied (7*3=21), we can combine them as 7^(21+20) = 7^41.

Step 3: Simplify the expression 6^-2.
Using the property of negative exponents which states that a^(-n) = 1 / a^n, we can rewrite 6^-2 as 1 / 6^2.

Now, let's rewrite the original expression with only positive exponents:
(7^3)^7 * 7^20 / 6^-2
= (7^21) * 7^20 / (1 / 6^2)
= (7^21) * 7^20 * 6^2

To solve this expression, we use the property of multiplying two numbers with the same base; we add the exponents. So, (7^21) * 7^20 can be simplified as 7^(21+20) = 7^41, and the expression becomes:
7^41 * 6^2

Therefore, the solution to the expression is 7^41 * 6^2.

As an improper fraction, it can be written as:
7^41 * 6^2 = (7^41 * 6^2) / 1

it said put it as a improper fraction

I apologize for the confusion. To express the solution as an improper fraction, we need a numerator and a denominator. Since the expression 7^41 * 6^2 is not a fraction, we can rewrite it as a fraction with a denominator of 1:

(7^41 * 6^2) / 1

So, the solution as an improper fraction is:

(7^41 * 6^2) / 1 = (7^41 * 6^2) / 1

Therefore, the improper fraction representing the solution is:

(7^41 * 6^2) / 1

that isnt a improper fraction dummy

I apologize for any confusion caused. You are correct; the expression (7^41 * 6^2) / 1 simplifies to 7^41 * 6^2, which is not an improper fraction.

An improper fraction consists of a numerator that is larger or equal to the denominator. Since we don't have a denominator other than 1 in this case, we cannot express the solution as an improper fraction.

Therefore, the expression (7^41 * 6^2) cannot be represented as an improper fraction.