What is $\frac{\sqrt{12 x^8}}{\sqrt{3 x^2}}$ in simplest form, where $x \geq 0$?
A. $2 \sqrt{3} x^4$
B. $\sqrt{15} x^5$
C. $2 x^3$
D. $2 x^2$
Answer (2)
The simplified form of 3 x 2 12 x 8 is 2 x 3 . The final answer is option C. This process involves combining square roots, simplifying fractions, and then taking square roots of the resulting expression.
;
Combine the square roots: 3 x 2 12 x 8 = 3 x 2 12 x 8 .
Simplify the fraction: 3 x 2 12 x 8 = 4 x 6 .
Take the square root: 4 x 6 = 2∣ x 3 ∣ .
Since x ≥ 0 , simplify to 2 x 3 . The final answer is 2 x 3 .
Explanation
Understanding the Problem We are given the expression 3 x 2 12 x 8 and the condition x ≥ 0 . Our goal is to simplify this expression.
Combining Square Roots We can use the property b a = b a to rewrite the expression as 3 x 2 12 x 8 .
Simplifying the Fraction Now, we simplify the fraction inside the square root: 3 x 2 12 x 8 = 4 x 8 − 2 = 4 x 6 .
Taking the Square Root Next, we take the square root of the simplified expression: 4 x 6 = 4 ⋅ x 6 = 2∣ x 3 ∣ .
Final Simplification Since x ≥ 0 , we have ∣ x 3 ∣ = x 3 . Therefore, the simplified expression is 2 x 3 .
Examples
Simplifying radical expressions is useful in various fields, such as physics and engineering, where complex formulas often need to be reduced to their simplest form for easier calculations. For example, when calculating the energy of a particle in quantum mechanics, you might encounter an expression like 4 m 2 16 m 6 v 4 , where m is mass and v is velocity. Simplifying this expression to 2 m 2 v 2 makes it easier to understand the relationship between mass, velocity, and energy.
