Simplify: [tex]$\sqrt{\frac{27 x^{12}}{300 x^8}}$[/tex]
Answer (1)
Simplify the fraction inside the square root: 300 27 = 100 9 .
Simplify the x terms using the quotient rule: x 8 x 12 = x 4 .
Take the square root of the simplified expression: 100 9 x 4 = 10 3 x 2 .
The simplified expression is 10 3 x 2 .
Explanation
Understanding the Problem We are asked to simplify the expression 300 x 8 27 x 12 . Let's break it down step by step.
Simplifying the Fraction First, simplify the fraction 300 27 by finding the greatest common divisor (GCD) of 27 and 300. The GCD of 27 and 300 is 3. Dividing both the numerator and the denominator by 3, we get: 300 27 = 300 ÷ 3 27 ÷ 3 = 100 9 So, the expression becomes 100 x 8 9 x 12 .
Simplifying the x Terms Next, simplify the x terms using the quotient rule for exponents, which states that x b x a = x a − b . In this case, we have: x 8 x 12 = x 12 − 8 = x 4 So, the expression becomes 100 9 x 4 .
Taking the Square Root Now, take the square root of the simplified expression. Recall that b a = b a and x 4 = x 2 . Thus, we have: 100 9 x 4 = 100 9 x 4 = 100 9 ⋅ x 4 = 10 3 x 2 So, the simplified expression is 10 3 x 2 .
Final Answer Therefore, the simplified expression is 10 3 x 2 .
Examples
Imagine you're designing a square garden and need to calculate its area. If the area is expressed as 300 x 8 27 x 12 , simplifying it to 10 3 x 2 helps you easily determine the actual area based on the value of x . This kind of simplification is useful in various fields like engineering, physics, and computer graphics, where complex expressions need to be made manageable for calculations and simulations.
