Simplify: [tex]$\sqrt{18 x^7}$[/tex]
Answer (2)
The expression 18 x 7 simplifies to 3 x 3 2 x by factoring out perfect squares from the constant and variable parts under the radical. By taking the square root of these perfect squares, we arrive at the simplified form easily. This technique is useful in various mathematical contexts, especially when simplifying equations.
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Factor the constant and variable parts of the expression under the radical.
Identify and extract perfect square factors.
Simplify the expression by taking the square roots of the perfect squares.
The simplified form of 18 x 7 is 3 x 3 2 x .
Explanation
Understanding the Problem We are asked to simplify the radical expression 18 x 7 . This involves finding perfect square factors within the radical and extracting them.
Factoring the Expression First, we factor the constant 18 into its prime factors: 18 = 2 ⋅ 3 2 . Then we rewrite x 7 as x 6 ⋅ x , which can be further expressed as ( x 3 ) 2 ⋅ x .
Rewriting the Expression Now we rewrite the original expression using these factors: 18 x 7 = 2 ⋅ 3 2 ⋅ x 6 ⋅ x = 3 2 ⋅ ( x 3 ) 2 ⋅ 2 x .
Simplifying the Radical Next, we take out the perfect squares from under the radical. The square root of 3 2 is 3, and the square root of ( x 3 ) 2 is x 3 . Thus, we have: 3 2 ⋅ ( x 3 ) 2 ⋅ 2 x = 3 x 3 2 x .
Final Answer Therefore, the simplified expression is 3 x 3 2 x .
Examples
Simplifying radical expressions is useful in various fields, such as physics and engineering, where you might need to calculate distances or areas involving square roots. For example, when calculating the length of the diagonal of a square with side length s , you get d = 2 s 2 = s 2 . Similarly, in circuit analysis, simplifying radicals can help in determining impedance or resonant frequencies. Understanding how to simplify these expressions allows for more accurate and efficient calculations in these practical applications.
