Find the simplified product $\sqrt{2 x^3} \cdot \sqrt{18 x^5}$
Answer (2)
The simplified product of 2 x 3 ⋅ 18 x 5 is 6 x 4 by combining the square roots, simplifying the expression inside, and taking the square root. This involves multiplying the coefficients and adding the exponents for terms with the same base. Finally, we apply the square root to both the coefficient and the variable expression.
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Combine the square roots using the property a ⋅ b = ab : 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: 36 x 8 = 6 x 4 .
The simplified product is 6 x 4 .
Explanation
Combining Square Roots We are given the expression 2 x 3 ⋅ 18 x 5 and we want to simplify it. We will use the property that a ⋅ b = ab to combine the two square roots.
Rewriting the Expression Using the property a ⋅ b = ab , we can rewrite the expression as 2 x 3 ⋅ 18 x 5 .
Simplifying Inside the Square Root Now, we simplify the expression inside the square root. We have 2 x 3 ⋅ 18 x 5 = ( 2 ⋅ 18 ) ⋅ ( x 3 ⋅ x 5 ) . Since 2 ⋅ 18 = 36 and x 3 ⋅ x 5 = x 3 + 5 = x 8 , we get 36 x 8 .
Taking the Square Root So, our expression becomes 36 x 8 . Now, we take the square root of this expression. We have 36 x 8 = 36 ⋅ x 8 . Since 36 = 6 and x 8 = x 8/2 = x 4 , we get 6 x 4 .
Final Answer Therefore, the simplified product is 6 x 4 .
Examples
Simplifying radical expressions is useful in various fields, such as physics and engineering, when dealing with quantities involving square roots. For example, when calculating the kinetic energy of an object, which is given by the formula K E = 2 1 m v 2 , where v is the velocity, if the velocity is expressed in terms of square roots, simplifying the expression can make further calculations easier. Also, in geometry, when finding the length of the diagonal of a rectangle or the distance between two points, simplifying radical expressions is often necessary.
