What is the simplified form of the following expression?
$7(\sqrt[3]{2 x})-3(\sqrt[3]{16 x})-3(\sqrt[3]{8 x})$
A. $-5(\sqrt[3]{2 x})$
B. $5(\sqrt[3]{x})$
C. $\sqrt[3]{2 x}-6(\sqrt[3]{x})$
D. $-(\sqrt[3]{2 x})-6(\sqrt[3]{x})$
Answer (1)
Simplify 3 16 x to 2 3 2 x .
Simplify 3 8 x to 2 3 x .
Substitute the simplified radicals back into the original expression and combine like terms.
The simplified expression is 3 2 x − 6 ( 3 x ) .
Explanation
Analyze the problem We are given the expression 7 ( 3 2 x ) − 3 ( 3 16 x ) − 3 ( 3 8 x ) and asked to simplify it. We will simplify each term and combine like terms to obtain the simplified expression.
Simplify the first radical First, we simplify the term 3 16 x . We can rewrite 16 as 8 ⋅ 2 , so we have 3 16 x = 3 8 ⋅ 2 x = 3 8 ⋅ 3 2 x = 2 3 2 x .
Simplify the second radical Next, we simplify the term 3 8 x . We know that 3 8 = 2 , so we have 3 8 x = 3 8 ⋅ 3 x = 2 3 x .
Substitute back into the expression Now we substitute these simplified radicals back into the original expression: 7 ( 3 2 x ) − 3 ( 3 16 x ) − 3 ( 3 8 x ) = 7 ( 3 2 x ) − 3 ( 2 3 2 x ) − 3 ( 2 3 x ) .
Combine like terms Now we simplify the expression by combining like terms: 7 ( 3 2 x ) − 3 ( 2 3 2 x ) − 3 ( 2 3 x ) = 7 ( 3 2 x ) − 6 ( 3 2 x ) − 6 ( 3 x ) = ( 7 − 6 ) ( 3 2 x ) − 6 ( 3 x ) = 3 2 x − 6 ( 3 x ) .
State the final answer Therefore, the simplified form of the given expression is 3 2 x − 6 ( 3 x ) .
Examples
Imagine you are designing storage containers, and you need to calculate the total volume required for different components. Some components have volumes expressed with cube roots. Simplifying expressions with radicals, like in this problem, allows you to efficiently combine and calculate the total volume needed, optimizing the design and minimizing material waste. This is crucial in engineering and design to ensure efficient use of space and resources.
