Question

Apply the Properties of Integer Exponents to generate equivalent expressions to (7^−3) ⋅ 7^20/6^−2
with only positive exponents, then solve the expression. Provide the answer as an improper fraction.(1 point)
(7^−3) ⋅ 7^20/6^−2=

Answer 1

now make into an improper fraction

Answer 2

To simplify the expression \((7^{-3}) \cdot 7^{20} / 6^{-2}\) using the properties of integer exponents, we can follow these steps:

1. **Combine the exponents of the base 7**:
Using the property \(a^m \cdot a^n = a^{m+n}\), we can combine \(7^{-3}\) and \(7^{20}\):

\[
7^{-3} \cdot 7^{20} = 7^{-3 + 20} = 7^{17}
\]

2. **Simplify the expression**:
Now, we rewrite the entire expression, substituting \(7^{17}\) for \((7^{-3}) \cdot 7^{20}\):

\[
\frac{7^{17}}{6^{-2}}
\]

3. **Convert the negative exponent in the denominator**:
Recall that \(\frac{1}{a^{-m}} = a^m\). Therefore, \(6^{-2}\) in the denominator can be written as:

\[
\frac{7^{17}}{6^{-2}} = 7^{17} \cdot 6^{2} = 7^{17} \cdot 6^{2}
\]

4. **Final expression**:
Now we have:

\[
7^{17} \cdot 6^{2}
\]

5. **Calculating numerical values**:
To calculate this, we can compute the individual components:

- \(6^2 = 36\)
- \(7^{17}\) is a larger number, and we will leave it in its exponential form or we can calculate it if needed.

Thus our expression is:

\[
7^{17} \cdot 36
\]

6. **Conclusion**:
The final result can be expressed as:

\[
\frac{7^{17} \cdot 36}{1}
\]

Therefore, the simplified expression with only positive exponents is:

\[
7^{17} \cdot 36
\]

Resulting in the answer as follows:

\[
\boxed{\frac{36 \cdot 7^{17}}{1}} \quad \text{or simply } \quad 36 \cdot 7^{17} \text{ as an improper fraction.}
\]

If you require \(7^{17}\)'s numerical approximation to present a specific fraction format, it might be needed, but generally, it can remain in exponent form.

Answer 3

To express the product \(36 \cdot 7^{17}\) as an improper fraction, we need to ensure it is in the form of \(\frac{a}{b}\), with \(a\) and \(b\) as integers and \(b\) not equal to zero.

We can write \(36 \cdot 7^{17}\) as:

\[
\frac{36 \cdot 7^{17}}{1}
\]

This representation is still an improper fraction since the numerator (which is \(36 \cdot 7^{17}\)) is larger than the denominator (which is 1).

If we want to keep \(36\) as part of a single expression, we can write:

\[
\frac{36 \cdot 7^{17}}{1}
\]

This is an improper fraction. To give a numerical value for \(7^{17}\), it is best expressed as follows:

Calculating \(7^{17}\):

\[
7^{17} = 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7
\]

Computing this might yield a very large number. Here are the actual values of \(7^{17}\) and \(36\):

\[
7^{17} = 23,834,668,764,117
\]

So,

\[
36 \cdot 7^{17} = 36 \cdot 23,834,668,764,117 = 858,420,108,299,212
\]

Thus, the improper fraction is:

\[
\frac{858,420,108,299,212}{1}
\]

Or simply:

\[
858,420,108,299,212
\]

So the final answer in an improper fraction format is:

\[
\boxed{\frac{36 \cdot 7^{17}}{1}} \quad \text{or }\quad \frac{858,420,108,299,212}{1}
\]

Both of these represent your answer correctly in improper fraction format.