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QuestionMathematics
Go step by step to reduce the radical:
Find the square root of 96.
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Answer
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Step 1:: Recognize that the given number, 96, is not a perfect square.
where $a$ is the number, $p_1, p_2, \ldots, p_n$ are prime factors, and $e_1, e_2, \ldots, e_n$ are their respective exponents.
To find its square root, we can use the following formula to successively extract any perfect squares within the number: \sqrt{a} = \sqrt{p_1^{e_1} p_2^{e_2} \ldots p_n^{e_n}} = p_1^{e_1 / 2} p_2^{e_2 / 2} \ldots p_n^{e_n/ 2}
Step 2:: Identify the prime factors of 96.
\begin{align*} 96 &= 2 \times 2 \times 2 \times 2 \times 2 \times 3 \ &= 2^5 \times 3 \end{align*}
Step 3:: Apply the formula to find the square root of 96.
\begin{align*} \sqrt{96} &= \sqrt{2^5 \times 3} \ &= \sqrt{2^2 \times 2^2 \times 2 \times 3} \ &= 2^{2 / 2} \times 2^{2 / 2} \times \sqrt{2 \times 3} \ &= 2 \times 2 \times \sqrt{6} \ &= 4\sqrt{6} \end{align*}
Final Answer
The square root of 96 is 4\sqrt{6}.
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