Go step by step to reduce the radical. \sqrt{162} \sqrt{2} \cdot \sqrt{81}
The Correct Answer and Explanation is:
The given expression involves multiple radicals:
1622⋅81\sqrt{162} \quad \sqrt{2} \cdot \sqrt{81}
To simplify step by step:
Step 1: Factorize 162
The number 162 can be expressed as:
162=2×81162 = 2 \times 81
Taking the square root on both sides:
162=2×81\sqrt{162} = \sqrt{2 \times 81}
Using the property of radicals:
a×b=a×b\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}
Applying this property:
162=2×81\sqrt{162} = \sqrt{2} \times \sqrt{81}
Step 2: Evaluate 81\sqrt{81}
Since 81 is a perfect square:
81=9\sqrt{81} = 9
Replacing this in the equation:
162=2×9\sqrt{162} = \sqrt{2} \times 9
Step 3: Evaluate 2⋅81\sqrt{2} \cdot \sqrt{81}
Using the previously determined value:
81=9\sqrt{81} = 9
Substituting in:
2⋅81=2⋅9\sqrt{2} \cdot \sqrt{81} = \sqrt{2} \cdot 9
Step 4: Final Answer
Both expressions simplify to:
929\sqrt{2}
Thus, the reduced form is:
929\sqrt{2}
Explanation
The approach follows a structured breakdown by factorizing numbers into their prime components. Recognizing perfect squares simplifies radical expressions efficiently. The property a×b=a×b\sqrt{a} \times \sqrt{b} = \sqrt{a \times b} plays a crucial role in breaking down complex square roots. When encountering numbers like 81, utilizing its perfect square nature allows extraction of integer values, reducing the complexity of the radical expression. The final form, 929\sqrt{2}, retains the non-perfect square part inside the radical while simplifying the coefficient outside. This method ensures accuracy and clarity in evaluating square roots systematically.
