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Multiple Choice Questions
Question 1 (1 point)
Write $6 \sqrt{5}$ as an entire radical.
$\sqrt{180}$
$\sqrt{30}$
$\sqrt{150}$
$\sqrt{900}$
Question 2 (1 point)
A cube has a volume of $7290 cm^3$. Determine the edge length of the cube as a radical in simplest form.
$81 \sqrt[3]{10} cm$
$9 \sqrt[3]{10} cm$
$9 \sqrt[3]{90} cm$
$10 \sqrt[3]{9} cm$
Answer (1)
To write 6 5 as an entire radical, square 6 and multiply by 5: 6 2 × 5 = 180 .
To find the edge length of a cube with volume 7290 cm 3 , take the cube root: s = 3 7290 .
Simplify the cube root: 3 7290 = 3 9 3 × 10 = 9 3 10 .
The edge length of the cube is 9 3 10 cm .
Explanation
Problem Analysis We will solve two multiple-choice questions related to radicals. The first question asks us to express 6 5 as an entire radical. The second question involves finding the edge length of a cube given its volume.
Question 1 Solution To express 6 5 as an entire radical, we need to move the coefficient 6 under the square root. To do this, we square the coefficient and multiply it by the radicand. So, we have: 6 5 = 6 2 × 5 = 36 × 5 = 180 Therefore, the entire radical is 180 .
Question 2 Solution For the second question, we are given that the volume of a cube is 7290 cm 3 . We need to find the edge length of the cube in simplest radical form. Let s be the edge length of the cube. The volume V of the cube is given by V = s 3 . Thus, we have: s 3 = 7290 To find the edge length s , we take the cube root of the volume: s = 3 7290 Now, we simplify the radical. First, find the prime factorization of 7290: 7290 = 2 × 3 6 × 5 = 2 × 729 × 5 = 9 3 × 2 × 5/9 = 3 6 ∗ 10 = 9 3 ∗ 10 So, we can write: s = 3 7290 = 3 9 3 × 10 = 9 3 10 Therefore, the edge length of the cube is 9 3 10 cm .
Final Answer For Question 1, the entire radical form of 6 5 is 180 .
For Question 2, the edge length of the cube with volume 7290 cm 3 is 9 3 10 cm .
Examples
Radicals are useful in various real-life situations, such as calculating distances, areas, and volumes. For example, if you are designing a square garden with an area of 150 square feet, the side length of the garden would be 150 feet. Simplifying this radical helps you determine the exact length needed for fencing or other materials. Similarly, in construction, radicals are used to calculate diagonal lengths or to ensure precise measurements in building structures.
