Apply the properties of integer exponents to generate equivalent expressions of (6 to the power of -3) to the power of 7 times 6 to the power of 20/6 to the power of -2. Answer simplified improper fraction. Even if you get a whole number answer.

Answer 1

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

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

2. Simplifying \(\frac{6^{20}}{6^{-2}}\): Using the quotient property, \(\frac{a^m}{a^n} = a^{m-n}\): \[ \frac{6^{20}}{6^{-2}} = 6^{20 - (-2)} = 6^{20 + 2} = 6^{22} \]

3. Now combine the two parts: We now have: \[ 6^{-21} \times 6^{22} \] Again using the product property, \(a^m \times a^n = a^{m+n}\): \[ 6^{-21 + 22} = 6^{1} = 6 \]

So, the expression simplifies to:

\[ 6 \]

Since the problem prompts for an improper fraction form, we can express 6 as:

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

Therefore, the final simplified expression is:

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