Question

Apply the Properties of Integer Exponents to simplify as a numerical value in IMPROPER fraction form [ (6^-3)^7 x 6^20 ] ÷ [6^-2]

Answer 1

Can you do it like an 8th grader

Answer 2

To simplify the expression \(\frac{(6^{-3})^7 \times 6^{20}}{6^{-2}}\), we can apply the properties of integer exponents step-by-step.

1. **Apply the power of a power property:**
\[
(a^m)^n = a^{m \cdot n}
\]
Thus,
\[
(6^{-3})^7 = 6^{-3 \cdot 7} = 6^{-21}
\]

2. **Substituting back into the expression:**
\[
\frac{6^{-21} \times 6^{20}}{6^{-2}}
\]

3. **Combine exponents in the numerator using the product of powers property:**
\[
a^m \times a^n = a^{m+n}
\]
So,
\[
6^{-21} \times 6^{20} = 6^{-21 + 20} = 6^{-1}
\]

4. **Now our expression simplifies to:**
\[
\frac{6^{-1}}{6^{-2}}
\]

5. **Apply the quotient of powers property:**
\[
\frac{a^m}{a^n} = a^{m-n}
\]
Thus,
\[
\frac{6^{-1}}{6^{-2}} = 6^{-1 - (-2)} = 6^{-1 + 2} = 6^{1} = 6
\]

6. **Final result:**
The simplified form of the expression \(\frac{(6^{-3})^7 \times 6^{20}}{6^{-2}}\) is:
\[
6
\]

To present this answer in improper fraction form:
\[
6 = \frac{6}{1}
\]

Thus, the final answer is:
\[
\boxed{\frac{6}{1}}
\]

Answer 3

Sure! Let’s break down the problem step-by-step like an 8th grader would learn it.

We need to simplify the expression:

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

### Step 1: Simplify \((6^{-3})^7\)

When we have a power raised to another power, we multiply the exponents. So:

\[
(6^{-3})^7 = 6^{-3 \times 7} = 6^{-21}.
\]

### Step 2: Substitute that back into the expression

Now our expression looks like this:

\[
\frac{6^{-21} \times 6^{20}}{6^{-2}}.
\]

### Step 3: Simplify the numerator \(6^{-21} \times 6^{20}\)

Using the rule that says when you multiply the same base, you add the exponents:

\[
6^{-21} \times 6^{20} = 6^{-21 + 20} = 6^{-1}.
\]

### Step 4: Now our expression is simpler:

It now looks like this:

\[
\frac{6^{-1}}{6^{-2}}.
\]

### Step 5: Simplify the fraction

When you divide two powers with the same base, you subtract the exponents:

\[
\frac{6^{-1}}{6^{-2}} = 6^{-1 - (-2)} = 6^{-1 + 2} = 6^{1} = 6.
\]

### Final Step: Write it in improper fraction form

So the simplified answer is:

\[
6.
\]

To express \(6\) as an improper fraction, we write it as:

\[
\frac{6}{1}.
\]

### Conclusion

The final answer is:

\[
\boxed{\frac{6}{1}}.
\]