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

3 answers

To simplify the expression \((7^{-3})^7 \cdot \frac{7^{20}}{6^{-2}}\) and generate equivalent expressions with only positive exponents, we can use the properties of integer exponents.

  1. Apply the power of a power property:

    \[ (7^{-3})^7 = 7^{-3 \cdot 7} = 7^{-21} \]

  2. Next, we rewrite the expression:

    \[ 7^{-21} \cdot \frac{7^{20}}{6^{-2}} \]

  3. Rewrite the denominator with positive exponents:

    \(6^{-2} = \frac{1}{6^2}\), therefore:

    \[ \frac{7^{20}}{6^{-2}} = 7^{20} \cdot 6^2 \]

    This allows us to rewrite the whole expression:

    \[ 7^{-21} \cdot 7^{20} \cdot 6^2 \]

  4. Combine the powers of \(7\):

    \[ 7^{-21 + 20} \cdot 6^2 = 7^{-1} \cdot 6^2 \]

  5. Changing \(7^{-1}\) to positive exponent:

    \[ 7^{-1} = \frac{1}{7} \]

    Therefore, the expression is:

    \[ \frac{1}{7} \cdot 6^2 \]

  6. Calculate \(6^2\):

    \[ 6^2 = 36 \]

    So, we have:

    \[ \frac{1}{7} \cdot 36 = \frac{36}{7} \]

Thus, the final answer is:

\[ \frac{36}{7} \]

This is the expression simplified with only positive exponents, and the answer in improper fraction form is:

\[ \boxed{\frac{36}{7}} \]

Bot, can you be more clear?

Sure! Let's break it down step-by-step to clarify the process of simplifying the expression \((7^{-3})^7 \cdot \frac{7^{20}}{6^{-2}}\) and ensuring that we only use positive exponents.

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

Using the power of a power property: \[ (7^{-3})^7 = 7^{-3 \cdot 7} = 7^{-21} \]

Step 2: Rewrite the Expression

Now we plug this back into our expression: \[ 7^{-21} \cdot \frac{7^{20}}{6^{-2}} \]

Step 3: Handle the Denominator

Recall that \(6^{-2}\) can be rewritten with a positive exponent: \[ 6^{-2} = \frac{1}{6^2} \] This means that dividing by \(6^{-2}\) is the same as multiplying by \(6^2\): \[ \frac{7^{20}}{6^{-2}} = 7^{20} \cdot 6^2 \]

Step 4: Combine Everything

Now substitute this back into our expression: \[ 7^{-21} \cdot 7^{20} \cdot 6^2 \]

Step 5: Combine Powers of \(7\)

Now we can combine the powers of \(7\): \[ 7^{-21 + 20} = 7^{-1} \] So the expression becomes: \[ 7^{-1} \cdot 6^2 \]

Step 6: Convert to Positive Exponents

Convert \(7^{-1}\) to a positive exponent: \[ 7^{-1} = \frac{1}{7} \]

Step 7: Calculate \(6^2\)

Now calculate \(6^2\): \[ 6^2 = 36 \]

Step 8: Combine All Parts

Putting it all together: \[ \frac{1}{7} \cdot 36 = \frac{36}{7} \]

Final Answer

The final expression is: \[ \boxed{\frac{36}{7}} \]

This is the result expressed as an improper fraction with all positive exponents. Let me know if you need further clarification!