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(sqrt(8c^(20)))/(sqrt(32c^(4))) Which of the following is equivalent to the given expression for all c!=0 ? Choose 1 answer: (A) (1)/(2)c^(4) (B) (1)/(2)c^(8) (C) (1)/(4)c^(16) (D) (1)/(4)c^(2)sqrtc

8c2032c4 \frac{\sqrt{8 c^{20}}}{\sqrt{32 c^{4}}} \newlineWhich of the following is equivalent to the given expression for all c≠0 c \neq 0 ?\newlineChoose 11 answer:\newline(A) 12c4 \frac{1}{2} c^{4} \newline(B) 12c8 \frac{1}{2} c^{8} \newline(C) 14c16 \frac{1}{4} c^{16} \newline(D) 14c2c \frac{1}{4} c^{2} \sqrt{c}

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Q. 8c2032c4 \frac{\sqrt{8 c^{20}}}{\sqrt{32 c^{4}}} \newlineWhich of the following is equivalent to the given expression for all c≠0 c \neq 0 ?\newlineChoose 11 answer:\newline(A) 12c4 \frac{1}{2} c^{4} \newline(B) 12c8 \frac{1}{2} c^{8} \newline(C) 14c16 \frac{1}{4} c^{16} \newline(D) 14c2c \frac{1}{4} c^{2} \sqrt{c}
  1. Simplify expression using properties: Simplify the given expression using the properties of radicals and exponents.\newlineThe given expression is 8c2032c4\frac{\sqrt{8c^{20}}}{\sqrt{32c^{4}}}.\newlineWe can simplify the square roots separately first.\newline8c20=23⋅c2⋅10=2⋅c10\sqrt{8c^{20}} = \sqrt{2^3 \cdot c^{2\cdot10}} = 2 \cdot c^{10} because a2=a\sqrt{a^2} = a for any positive real number aa.\newline32c4=25⋅c2⋅2=4⋅c2\sqrt{32c^{4}} = \sqrt{2^5 \cdot c^{2\cdot2}} = 4 \cdot c^{2} because a2=a\sqrt{a^2} = a for any positive real number aa.
  2. Simplify square roots separately: Divide the simplified square roots.\newlineNow we divide (2⋅c10)(2 \cdot c^{10}) by (4⋅c2)(4 \cdot c^{2}).\newline(2⋅c10)/(4⋅c2)=(2/4)⋅c(10−2)=(1/2)⋅c8(2 \cdot c^{10}) / (4 \cdot c^{2}) = (2/4) \cdot c^{(10-2)} = (1/2) \cdot c^{8}.
  3. Divide simplified square roots: Check the answer choices to see which one matches our simplified expression.\newlineThe expression we found is (12)⋅c8(\frac{1}{2}) \cdot c^{8}, which matches answer choice (B).

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