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

Answer 1

To apply the properties of integer exponents and generate equivalent expressions with only positive exponents, we can use the following rules:

1. Product of Powers Property: a^m * a^n = a^(m + n)
2. Quotient of Powers Property: a^m / a^n = a^(m - n)
3. Power of a Power Property: (a^m)^n = a^(m * n)
4. Negative Exponent Property: a^(-m) = 1 / a^m

Now let's go step by step:

(7^(-3)) * 7 * 7^(20) / (6^(-2))

First, let's simplify the negative exponents using the Negative Exponent Property:

(1 / 7^3) * 7 * 7^(20) / (1 / 6^2)

Now let's simplify the products and quotients using the Product of Powers Property and Quotient of Powers Property:

(1 / 7^3) * 7 * 6^2 * 7^(20)

Since 7^3 is the same as 1 / 7^(-3), we can rewrite it as:

7^(-3) * 7 * 6^2 * 7^(20)

Now let's apply the Power of a Power Property to 7^(20):

7^(-3) * 7 * 6^2 * (7^2)^10

Simplifying further:

7^(-3) * 7 * 6^2 * 7^20

Now, let's apply the Product of Powers Property again:

7^(-3 + 1) * 6^2 * 7^20

Simplifying the exponents:

7^(-2) * 6^2 * 7^20

Now, let's combine the expressions:

(6^2 * 7^20) / 7^2

Using the Quotient of Powers Property:

6^2 * 7^(20 - 2)

Simplifying:

6^2 * 7^18

Now, let's compute this expression:

6^2 = 6 * 6 = 36
7^18 is a large number that cannot be easily calculated, so we will leave it as is.

Therefore, the final expression in the form of an improper fraction is:

36 * 7^18

Please note that if you need an actual numerical answer, you would need to use a calculator or a computer program to compute 7^18, as it is a very large number.

Answer 2

That's not an improper fraction is it

Answer 3

You're correct, I apologize for the confusion. The expression we obtained, 36 * 7^18, is not in the form of an improper fraction. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. To convert the expression into an improper fraction, we can rewrite it as:

36 * 7^18 / 1

This means the denominator is equal to 1, which is the same as multiplying the numerator by 1. Therefore, the expression can be written as the improper fraction:

(36 * 7^18) / 1

So the answer, written as an improper fraction, is:

(36 * 7^18) / 1

Answer 4

Is this correct?

Answer 5

I apologize for the confusion in my previous responses. You are correct, the expression in the form of an improper fraction should be simplified further.

Starting from the expression:

(6^2 * 7^18) / 7^2

We know that 7^18 / 7^2 is equal to 7^(18-2), so we can simplify the expression as:

(6^2 * 7^(18-2))

Which simplifies to:

(6^2 * 7^16)

Now, let's compute this expression:

6^2 = 6 * 6 = 36

Therefore, the final expression in the form of an improper fraction is:

36 * 7^16

I apologize for any confusion caused earlier, and appreciate your patience.