Question

Apply the properties of integer exponents to generate equivalent expressions to (7-power of three) then power of 7×7 power of 20/6 minus power of two positive exponents then solve the expression provide the answer as an improper fraction

Answer 1

To simplify the expression you provided, we need to rewrite it properly using the properties of integer exponents.

The expression seems to be:

\[
(7^3)^{7^{20}} / 6 - 2
\]

Now let's apply the properties of exponents.

1. First, simplify \( (7^3)^{7^{20}} \).

Using the power of a power property, which states that \( (a^m)^n = a^{m \cdot n} \), we get:

\[
(7^3)^{7^{20}} = 7^{3 \cdot 7^{20}}
\]

2. The goal now is to simplify \( 7^{3 \cdot 7^{20}} \):

- We can manipulate the exponents further. Since \( 7^{20} = 7 \times 7^{19} \), we can multiply the exponents:

\[
3 \cdot 7^{20} = 3 \times 7^{20}
\]

Thus, we can express the number simply as \( 7^{(3 \cdot 7^{20})} \).

3. To address the division by 6, we can rewrite our expression:

\[
\frac{7^{(3 \cdot 7^{20})}}{6} - 2
\]

But we want to express the answer as an improper fraction, we have \( \frac{7^{(3 \cdot 7^{20})}}{6} \).

To show that the resulting expression minus 2 can also be expressed as an improper fraction, we will represent 2 as a fraction:

\[
2 = \frac{12}{6}
\]

Thus, our entire expression becomes:

\[
\frac{7^{(3 \cdot 7^{20})}}{6} - \frac{12}{6} = \frac{7^{(3 \cdot 7^{20})} - 12}{6}
\]

Now the expression has been converted to an improper fraction format.

In the simplified form, we find that the numerator is \( 7^{(3 \cdot 7^{20})} - 12 \).

Since \( 7^{(3 \cdot 7^{20})} \) is a huge value from the exponentiation, it definitely gives a very large integer.

Thus, the answer in improper fraction form to your original expression can be given as:

\[
\frac{7^{(3 \cdot 7^{20})} - 12}{6}
\]

This format now represents the answer as an improper fraction, with the numerator being \( 7^{(3 \cdot 7^{20})} - 12 \) and the denominator 6.