Which expression is equivalent to $\sqrt[4]{144 a^{12} b^3}$ ? Assume $a \geq 0$ and $b \geq 0$.
A. $2 a^3(\sqrt[4]{9 b^3})$
B. $2 a^4 b(\sqrt[4]{18})$
C. $12 a^3(\sqrt[4]{b^3})$
D. $12 a^6 b(\sqrt[4]{b})$
Answer (2)
The equivalent expression for 4 144 a 12 b 3 is 2 a 3 ( 4 9 b 3 ) , which corresponds to option A. The simplification involved factoring the constant and separating the fourth roots for each term. Overall, this leads to the solution provided in the options.
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Simplify the constant term inside the radical: 144 = 2 4 × 3 2 .
Separate the fourth root into individual terms: 4 2 4 × 3 2 × a 12 × b 3 = 4 2 4 × 4 3 2 × 4 a 12 × 4 b 3 .
Simplify each term: 4 2 4 = 2 , 4 a 12 = a 3 , and rewrite 4 3 2 as 4 9 .
Combine the terms to get the final answer: 2 a 3 ( 4 9 b 3 ) .
Explanation
Understanding the Problem We are given the expression 4 144 a 12 b 3 with the conditions a ≥ 0 and b ≥ 0 . We want to find an equivalent expression from the given options. We will use properties of radicals and exponents to simplify the given expression.
Simplifying the Constant Term First, let's simplify the constant term inside the radical. We know that 144 = 16 × 9 = 2 4 × 3 2 . So we can rewrite the expression as 4 2 4 × 3 2 × a 12 × b 3 .
Separating the Fourth Root Now, we can separate the fourth root into individual terms: 4 2 4 × 4 3 2 × 4 a 12 × 4 b 3 .
Simplifying Each Term Simplify each term: 4 2 4 = 2 , 4 a 12 = a 12/4 = a 3 . The term 4 3 2 can be written as 4 9 , and 4 b 3 remains as is. Thus, we have 2 × 4 9 × a 3 × 4 b 3 .
Combining the Terms Combining the terms, we get 2 a 3 4 9 b 3 .
Final Answer Comparing this with the given options, we see that the equivalent expression is 2 a 3 ( 4 9 b 3 ) .
Examples
Radicals and exponents are used in various fields such as physics, engineering, and computer graphics. For example, in physics, the period of a pendulum can be expressed using a square root. In computer graphics, transformations such as scaling and rotation often involve radicals and exponents. Understanding how to simplify radical expressions can help in solving problems in these areas.
