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(7^-3)^7 • 7^20/ 6^-2

5 answers

To simplify the expression \((7^{-3})^7 \cdot 7^{20} / 6^{-2}\), we can follow these steps:

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

  2. Combine the powers of \(7\): Now we have: \[ 7^{-21} \cdot 7^{20} = 7^{-21 + 20} = 7^{-1} \]

  3. Combine with \(6^{-2}\): The expression simplifies to: \[ \frac{7^{-1}}{6^{-2}} = 7^{-1} \cdot 6^{2} \]

  4. Rewrite in fraction form: \[ 7^{-1} \cdot 6^{2} = \frac{6^{2}}{7^{1}} = \frac{6^2}{7} = \frac{36}{7} \]

Thus, the simplified expression is: \[ \boxed{\frac{36}{7}} \]

Can u explain more

Certainly! Let's break down the expression \((7^{-3})^7 \cdot 7^{20} / 6^{-2}\) step by step, explaining each part in detail.

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

  • Exponentiation Rule: When you raise a power to another power, you multiply the exponents. Mathematically, this can be expressed as: \[ (a^m)^n = a^{m \cdot n} \]
  • Applying this rule here: \[ (7^{-3})^7 = 7^{-3 \cdot 7} = 7^{-21} \]

Step 2: Combine with \(7^{20}\)

  • We now have: \[ 7^{-21} \cdot 7^{20} \]
  • Multiplying with the same base: When you multiply terms with the same base, you add the exponents: \[ a^m \cdot a^n = a^{m+n} \]
  • Using this rule: \[ 7^{-21} \cdot 7^{20} = 7^{-21 + 20} = 7^{-1} \]

Step 3: Simplifying the Division by \(6^{-2}\)

  • Next, we have: \[ \frac{7^{-1}}{6^{-2}} \]
  • Understanding Negative Exponents: A negative exponent indicates the reciprocal: \[ a^{-m} = \frac{1}{a^m} \]
  • Rewriting \(6^{-2}\) using the reciprocal: \[ 6^{-2} = \frac{1}{6^2} \]
  • Therefore, the division can be rewritten: \[ \frac{7^{-1}}{6^{-2}} = 7^{-1} \cdot 6^{2} \]

Step 4: Simplifying \(6^{2}\) and Final Combination

  • Expressing with Positive Exponents: Since \(7^{-1} = \frac{1}{7}\): \[ 7^{-1} \cdot 6^{2} = \frac{6^2}{7} \]
  • Now calculate \(6^2\): \[ 6^2 = 36 \]
  • So we have: \[ \frac{36}{7} \]

Conclusion:

Putting all the steps together, the entire expression simplifies down to: \[ \frac{36}{7} \] Thus, the final answer is: \[ \boxed{\frac{36}{7}} \]

Can u put it in number form is it a fraction is it 36/7?

Yes, the expression simplifies to a fraction, and in number form, it is indeed \(\frac{36}{7}\).

To convert this fraction to a mixed number or a decimal:

  1. Mixed Number: \(\frac{36}{7}\) can be expressed as a mixed number.

    • \(36\) divided by \(7\) is \(5\) with a remainder of \(1\).
    • So, \(\frac{36}{7} = 5 \frac{1}{7}\).

  2. Decimal: To express \(\frac{36}{7}\) as a decimal, you perform the division: \[ 36 \div 7 \approx 5.142857... \] The decimal \(5.142857\) has a repeating part (the "142857" repeats).

So, in number form:

  • As a fraction: \(\frac{36}{7}\)
  • As a mixed number: \(5 \frac{1}{7}\)
  • As a decimal: \(5.142857...\)

If you need one specific form, feel free to let me know!