Which expression equals $9 \sqrt[3]{10}$ ?
A. $5 \sqrt{10}+4 \sqrt{10}$
B. $5 \sqrt[3]{10}+4 \sqrt[3]{10}$
C. $5 \sqrt{10}+4 \sqrt[3]{10}$
D. $5 \sqrt[3]{10}+4 \sqrt{10}$
Answer (1)
$\boxed{5 3 10
4 3 10 }$
Explanation
Understanding the Problem We are given the expression $9 3 10 and four other expressions. Our goal is to determine which of the four expressions is equal to $9 3 10 .
Solution Plan Let's simplify each of the four given expressions and compare them with $9 3 10 to find the matching expression.
Simplifying Expression 1 Expression 1: $5 10
4 10 = (5+4) 10 = 9 10
Simplifying Expression 2 Expression 2: $5 3 10
4 3 10 = (5+4) 3 10 = 9 3 10
Simplifying Expression 3 Expression 3: $5 10
4 3 10 (cannot be simplified further)
Simplifying Expression 4 Expression 4: $5 3 10
4 10 (cannot be simplified further)
Finding the Matching Expression Comparing each simplified expression to $9 3 10 , we find that the second expression, $5 3 10
4 3 10 , simplifies to $9 3 10 , which matches the given expression.
Conclusion Therefore, the expression that equals $9 3 10 is $5 3 10
4 3 10 .
Examples
Radicals and roots are not just abstract math concepts; they appear in various real-world applications. For instance, when calculating the period of a pendulum, you use a square root. If you're designing a cubic container and need to determine the side length based on a given volume, you would use a cube root. Understanding how to simplify and combine radical expressions is crucial in fields like physics, engineering, and computer graphics, where these calculations are commonplace.
