Which expression is equivalent to $\sqrt[3]{64 a^6 b^7 c^9}$?
A. $2 a b c^2(\sqrt[3]{4 a^2 b^3 c})$
B. $4 a^2 b^2 c^3(\sqrt[3]{b})$
C. $8 a^3 b^3 c^4(\sqrt[3]{b c})$
D. $8 a^2 b^2 c^3(\sqrt[3]{b})$
Answer (2)
Simplify the cube root of each factor in the expression: 3 64 , 3 a 6 , 3 b 7 , and 3 c 9 .
Rewrite b 7 as b 6 ⋅ b to simplify its cube root.
Calculate the cube roots: 3 64 = 4 , 3 a 6 = a 2 , 3 b 6 ⋅ b = b 2 3 b , and 3 c 9 = c 3 .
Combine the simplified terms to get the final expression: 4 a 2 b 2 c 3 ( 3 b ) .
Explanation
Understanding the Problem We are given the expression 3 64 a 6 b 7 c 9 and asked to find an equivalent expression from the given options.
Strategy To simplify the given expression, we take the cube root of each factor. Recall that n x m = x n m .
Simplifying the Expression We can rewrite the expression as follows: 3 64 a 6 b 7 c 9 = 3 64 ⋅ 3 a 6 ⋅ 3 b 7 ⋅ 3 c 9 We know that 3 64 = 4 since 4 3 = 64 . Also, 3 a 6 = a 3 6 = a 2 and 3 c 9 = c 3 9 = c 3 .
For the term b 7 , we can rewrite it as b 7 = b 6 + 1 = b 6 ⋅ b . Thus, 3 b 7 = 3 b 6 ⋅ b = 3 b 6 ⋅ 3 b = b 3 6 ⋅ 3 b = b 2 3 b .
Combining the Terms Substituting these values back into the expression, we get: 3 64 a 6 b 7 c 9 = 4 ⋅ a 2 ⋅ b 2 3 b ⋅ c 3 = 4 a 2 b 2 c 3 3 b .
Final Answer Comparing this simplified expression with the given options, we see that the equivalent expression is 4 a 2 b 2 c 3 ( 3 b ) .
Conclusion Therefore, the equivalent expression is 4 a 2 b 2 c 3 ( 3 b ) .
Examples
Understanding how to simplify radical expressions is useful in many areas of mathematics and science. For example, in physics, you might encounter such expressions when calculating the energy levels of a quantum mechanical system or when dealing with wave functions. In engineering, simplifying radicals can help in designing structures or analyzing circuits. Simplifying radicals allows for easier manipulation and understanding of complex equations, making problem-solving more efficient and accurate. For instance, if you are calculating the volume of a cube with side length 3 8 x 6 y 9 , simplifying this to 2 x 2 y 3 makes further calculations much easier.
The expression 3 64 a 6 b 7 c 9 simplifies to 4 a 2 b 2 c 3 3 b . The correct equivalent expression, after adjustment mistakes considered, is given correctly as option D: 8 a 2 b 2 c 3 ( 3 b ) .
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