Find the simplified product: [tex]\sqrt{2 x^3} \cdot \sqrt{18 x^5}[/tex]
Answer (2)
The simplified product of 2 x 3 ⋅ 18 x 5 is 6 x 4 . This is achieved by multiplying the terms under the square roots, simplifying the resulting expression, and evaluating the square roots. The final answer is derived from combining the results of the square roots.
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Multiply the terms under the square roots: 2 x 3 ⋅ 18 x 5 = 36 x 8 .
Simplify the square root: 36 x 8 = 36 ⋅ x 8 .
Evaluate the square roots: 36 = 6 and x 8 = x 4 .
Combine the terms to get the final simplified expression: 6 x 4 .
Explanation
Understanding the Problem We are given the expression 2 x 3 ⋅ 18 x 5 and we want to simplify it.
Multiplying Under the Square Root First, we multiply the terms under the square roots: 2 x 3 ⋅ 18 x 5 = 2 x 3 ⋅ 18 x 5 = 36 x 8 .
Separating the Square Root Next, we simplify the square root by taking the square root of 36 and x 8 separately: 36 x 8 = 36 ⋅ x 8 .
Evaluating Square Roots Now, we evaluate the square roots: 36 = 6 and x 8 = x 4 .
Combining Terms Finally, we combine the terms to get the simplified expression: 2 x 3 ⋅ 18 x 5 = 6 x 4 .
Examples
Understanding how to simplify radical expressions is useful in many areas of mathematics and physics. For example, when calculating the energy of a quantum harmonic oscillator, you often encounter expressions involving square roots and powers. Simplifying these expressions allows for easier manipulation and calculation of the energy levels.
