apply the properties of integer exponents to generate equivalent expressions of 6^-3^7x6^20 6^-2 answer as a simplified improper fraction even if you get a whole number answer this is a exponent fraction problem the line is a fraction line

Answer 1

To simplify the expression \( \frac{6^{-3} \cdot 6^{20}}{6^{-2}} \) using the properties of exponents, we can apply the rules of exponents:

1. Product of Powers: \( a^m \cdot a^n = a^{m+n} \)
2. Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \)

Let's start simplifying the expression step by step.

Step 1: Simplify the numerator

We first simplify the expression in the numerator:

\[ 6^{-3} \cdot 6^{20} = 6^{-3 + 20} = 6^{17} \]

Step 2: Write the expression as a single fraction

Now, we can rewrite the expression as follows:

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

Step 3: Simplify the fraction

Using the quotient of powers rule:

\[ 6^{17 - (-2)} = 6^{17 + 2} = 6^{19} \]

Step 4: Convert to an improper fraction

Since \( 6^{19} \) is a power of 6, we express it as an improper fraction:

\[ 6^{19} = \frac{6^{19}}{1} \]

Thus, the final simplified improper fraction expression is:

\[ \boxed{\frac{6^{19}}{1}} \]

This simplifies to \( 6^{19} \), but in terms of an improper fraction, it is \( \frac{6^{19}}{1} \). If needed, we could also calculate \( 6^{19} \) numerically, but it is often sufficient to leave it in its exponentiated form for simplicity.