Find the simplified product: [tex]$\sqrt{2 x^3} \cdot \sqrt{18 x^5}$[/tex]
Answer (2)
The product of 2 x 3 ⋅ 18 x 5 simplifies to 6 x 4 by combining the terms under a single square root, multiplying, and then taking the square root of the resulting expression. This involves basic operations of arithmetic and power rules. Understanding how to handle square roots is important in simplifying expressions effectively.
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Multiply the terms under the square roots: 2 x 3 ⋅ 18 x 5 = ( 2 x 3 ) ( 18 x 5 ) .
Simplify the expression inside the square root: ( 2 x 3 ) ( 18 x 5 ) = 36 x 8 .
Take the square root of the simplified expression: 36 x 8 = 36 ⋅ x 8 .
Simplify the square roots to get the final answer: 6 x 4 .
Explanation
Understanding the Problem We are given the expression 2 x 3 ⋅ 18 x 5 and asked to simplify it.
Multiplying Under the Square Roots To simplify the expression, we first multiply the terms under the square roots: 2 x 3 ⋅ 18 x 5 = ( 2 x 3 ) ( 18 x 5 )
Simplifying the Expression Next, we simplify the expression inside the square root: ( 2 x 3 ) ( 18 x 5 ) = 2 ⋅ 18 ⋅ x 3 ⋅ x 5 = 36 x 3 + 5 = 36 x 8 So we have: 36 x 8
Taking the Square Root Now, we take the square root of the simplified expression: 36 x 8 = 36 ⋅ x 8
Simplifying Further We simplify the square roots: 36 = 6 x 8 = x 4
Final Answer Finally, we combine the simplified terms to get the final answer: 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 physical quantities.
