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Apply the Properties of Integer Exponents to generate equivalent expressions to (7−3)7 ⋅ 7206−2

with only positive exponents, then solve the expression. Provide the answer as an improper fraction.

7 answers

To simplify the expression \( (7^{-3})^7 \cdot 7^{206 - 2} \), we will apply the properties of integer exponents.

Step 1: Simplify each part using the properties of exponents

  1. For \( (7^{-3})^7 \): \[ (7^{-3})^7 = 7^{-3 \cdot 7} = 7^{-21} \]

  2. For \( 7^{206 - 2} \): \[ 7^{206 - 2} = 7^{204} \]

Step 2: Combine the two parts

Now we combine these results: \[ 7^{-21} \cdot 7^{204} \]

Using the property of exponents \( a^m \cdot a^n = a^{m+n} \): \[ 7^{-21 + 204} = 7^{183} \]

Step 3: Convert the expression into positive exponents

\( 7^{183} \) is already expressed with a positive exponent.

Step 4: Solve the expression

The final expression \( 7^{183} \) is already simplified. However, if you need an improper fraction, you can write:

\[ 7^{183} = \frac{7^{183}}{1} \]

Thus, the answer, expressed as an improper fraction, is:

\[ \frac{7^{183}}{1} \]

The answer is: \[ \frac{7^{183}}{1} \]

This is the simplified improper fraction form of the evaluated expression.

can you please show the answer in a form i can understand

Certainly! Let's break it down step by step in a clearer format.

Given Expression

We start with the expression: \[ (7^{-3})^7 \cdot 7^{206 - 2} \]

Step 1: Simplify Each Part Using Properties of Exponents

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

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

  2. Simplify \( 7^{206 - 2} \):

    • Calculate \( 206 - 2 = 204\): \[ 7^{206 - 2} = 7^{204} \]

Step 2: Combine the Two Parts

Now we combine \( 7^{-21} \) and \( 7^{204} \) using the property of exponents \( a^m \cdot a^n = a^{m+n} \): \[ 7^{-21} \cdot 7^{204} = 7^{-21 + 204} = 7^{183} \]

Step 3: Present with Positive Exponents

The final expression \( 7^{183} \) is already in positive exponent form.

Final Answer

  • The answer can also be written as an improper fraction: \[ \frac{7^{183}}{1} \]

So the final answer is simply: \[ \frac{7^{183}}{1} \quad \text{(which is just } 7^{183} \text{)} \]

If you need the numerical value of \( 7^{183} \), it would be much larger and difficult to write out, but as an expression, it's best left as \( 7^{183} \) or \( \frac{7^{183}}{1} \). Let me know if you need further clarification!

instead of using code please put the math in a form i can understand (like using 1/4 for one fourth)

Sure! Let's break it down step by step in plain text without any code formatting.

Step-by-Step Solution

  1. Start with the expression: (7 to the power of -3) raised to the power of 7, multiplied by 7 raised to the power of (206 minus 2).

    Symbolically: (7^-3)^7 ⋅ 7^(206 - 2)

  2. Simplify (7^-3)^7:

    • When we raise a power to another power, we multiply the exponents.
    • So, (7^-3)^7 = 7^(-3 × 7) = 7^(-21).

  3. Simplify 7^(206 - 2):

    • First, calculate (206 - 2), which is 204.
    • Therefore, 7^(206 - 2) = 7^204.

  4. Combine the results: Now we multiply 7^(-21) by 7^204.

    • Using the rule that says a^m multiplied by a^n equals a^(m + n), we have: 7^(-21) × 7^204 = 7^(-21 + 204) = 7^(183).

  5. Final Result: The final expression is 7^(183).

If you need to express this in terms of a fraction with only positive exponents, you can write it as:

7^(183) = 7^(183) / 1.

Conclusion:

So the answer in a simple format is:

Final Answer: 7^(183), or as a fraction: 7^(183) / 1.

This represents the simplified expression with positive exponents. If you have any further questions or if something isn't clear, feel free to ask!

now you know my math language, thanks bot

You're welcome! I'm glad we could find a way to communicate effectively about the math. If you have any more questions or if there's anything else you'd like to discuss—math-related or otherwise—just let me know. I'm here to help!