$\sqrt[12]{\left(x^4\right)^{\frac{1}{3}}}$
Answer (1)
Rewrite the radical using fractional exponents: 12 ( x 4 ) 3 1 = ( ( x 4 ) 3 1 ) 12 1 .
Apply the power of a power rule: ( ( x 4 ) 3 1 ) 12 1 = x 4 ⋅ 3 1 ⋅ 12 1 .
Simplify the exponent: 4 ⋅ 3 1 ⋅ 12 1 = 9 1 .
Rewrite the expression in radical notation: x 9 1 = 9 x . The final answer is 9 x .
Explanation
Understanding the Problem We are given the expression 12 ( x 4 ) 3 1 . Our goal is to simplify this expression using the properties of exponents and radicals. Let's break it down step by step.
Rewriting with Fractional Exponents First, let's rewrite the radical using fractional exponents. Recall that n a = a n 1 . Therefore, we can rewrite the given expression as: 12 ( x 4 ) 3 1 = ( ( x 4 ) 3 1 ) 12 1 .
Applying the Power of a Power Rule Now, we use the power of a power rule, which states that ( a m ) n = a mn . Applying this rule, we get: ( ( x 4 ) 3 1 ) 12 1 = x 4 ⋅ 3 1 ⋅ 12 1 .
Simplifying the Exponent Next, we simplify the exponent by multiplying the fractions: 4 ⋅ 3 1 ⋅ 12 1 = 36 4 = 9 1 .So, our expression becomes: x 9 1 .
Converting Back to Radical Notation Finally, we rewrite the expression back in radical notation. Since x n 1 = n x , we have: x 9 1 = 9 x .Therefore, the simplified expression is 9 x .
Final Answer Thus, the simplified form of the given expression is 9 x .
Examples
Imagine you are calculating the growth of a plant. If the plant's growth can be modeled by the expression 12 ( x 4 ) 3 1 , where x represents some initial growth factor, simplifying this expression to 9 x allows you to more easily understand and calculate the plant's overall growth. This simplification makes it easier to compare the growth to other plants or to analyze the effects of different growth conditions.
