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Simplify: sqrt((82 4^(n+1))/(2^(2n) 16))

Mathematics

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Answer

4\textbf{4}

Step-by-step solution

6 steps

  1. Express all numbers in the expression as powers of 2.

    We know that 8=238 = 2^3, 4=224 = 2^2, and 16=2416 = 2^4.
    Substitute these into the expression:

    (23)2×(22)n+122n×24\sqrt{\frac{(2^3)^2 \times (2^2)^{n+1}}{2^{2n} \times 2^4}}

  2. Apply the exponent rule (am)n=amn(a^m)^n = a^{mn} to simplify the powers.

    23×2×22(n+1)22n×24=26×22n+222n×24\sqrt{\frac{2^{3 \times 2} \times 2^{2(n+1)}}{2^{2n} \times 2^4}} = \sqrt{\frac{2^6 \times 2^{2n+2}}{2^{2n} \times 2^4}}

  3. Apply the exponent rule am×an=am+na^m \times a^n = a^{m+n} to combine terms in the numerator and denominator.

    26+(2n+2)22n+4=22n+822n+4\sqrt{\frac{2^{6 + (2n+2)}}{2^{2n+4}}} = \sqrt{\frac{2^{2n+8}}{2^{2n+4}}}

  4. Apply the exponent rule aman=am−n\frac{a^m}{a^n} = a^{m-n} to simplify the fraction inside the square root.

    2(2n+8)−(2n+4)=22n+8−2n−4=24\sqrt{2^{(2n+8) - (2n+4)}} = \sqrt{2^{2n+8-2n-4}} = \sqrt{2^4}

  5. Evaluate the square root.

    24=(24)12=24×12=22\sqrt{2^4} = (2^4)^{\frac{1}{2}} = 2^{4 \times \frac{1}{2}} = 2^2

  6. Calculate the final value.

    22=42^2 = 4

    The simplified expression is:

    4\textbf{4}
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