Order these numbers from least to greatest.
[tex]\sqrt[3]{75}, \sqrt{14}, \sqrt[3]{100}, \sqrt{17}, \sqrt[3]{30}[/tex]
A. [tex]\sqrt[3]{75}, \sqrt[3]{100}, \sqrt{14}, \sqrt[3]{30}, \sqrt{17}[/tex]
B. [tex]\sqrt[3]{30}, \sqrt{14}, \sqrt{17}, \sqrt[3]{75}, \sqrt[3]{100}[/tex]
C. [tex]\sqrt[3]{100}, \sqrt[3]{30}, \sqrt{14}, \sqrt{17}, \sqrt[3]{75}[/tex]
D. [tex]\sqrt{17}, \sqrt[3]{75}, \sqrt[3]{100}, \sqrt{14}, \sqrt[3]{30}[/tex]
Simplify the radical [tex]\sqrt{98}[/tex].
A. [tex]7 \sqrt{2}[/tex]
B. [tex]7 \sqrt{14}[/tex]
C. [tex]49 \sqrt{2}[/tex]
D. [tex]2 \sqrt{7}[/tex]
Answer (2)
Approximate the given numbers: 3 30 ≈ 3.107 , 14 ≈ 3.742 , 17 ≈ 4.123 , 3 75 ≈ 4.217 , 3 100 ≈ 4.642 .
Order the numbers from least to greatest: 3 30 , 14 , 17 , 3 75 , 3 100 .
Simplify the radical: 98 = 2 ⋅ 7 2 = 7 2 .
The simplified radical and the ordered list of numbers are the final answers: 3 30 , 14 , 17 , 3 75 , 3 100 ; 7 2 .
Explanation
Problem Analysis We need to order the numbers 3 75 , 14 , 3 100 , 17 , 3 30 from least to greatest and simplify the radical 98 .
Ordering the Numbers First, let's approximate the values of the given numbers to compare them easily. We have:
3 30 ≈ 3.107 14 ≈ 3.742 17 ≈ 4.123 3 75 ≈ 4.217 3 100 ≈ 4.642
Therefore, the order from least to greatest is 3 30 , 14 , 17 , 3 75 , 3 100 .
Simplifying the Radical Now, let's simplify the radical 98 . We can find the prime factorization of 98 as 98 = 2 ⋅ 49 = 2 ⋅ 7 2 . Therefore,
98 = 2 ⋅ 7 2 = 7 2 ⋅ 2 = 7 2
Examples
Understanding how to order different types of radicals (square roots, cube roots, etc.) is useful in various real-world scenarios. For example, in construction, you might need to compare the lengths of materials given in radical form to ensure you're using the correct sizes. Similarly, in physics, comparing velocities or distances that involve radicals can help determine which object is moving faster or farther.
The ordered list of numbers from least to greatest is 3 30 , 14 , 17 , 3 75 , 3 100 , and the simplified form of 98 is 7 2 .
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