Rewrite $\sqrt[5]{4} \cdot \sqrt{2}$ as a single radical.
Answer (2)
To express 5 4 ⋅ 2 as a single radical, we first rewrite each radical as a power of 2, leading to the expression 2 10 9 . Finally, we can write this as 10 2 9 . Thus, the answer is 10 2 9 .
;
Rewrite 5 4 as 2 5 2 .
Rewrite 2 as 2 2 1 .
Combine the terms: 2 5 2 ⋅ 2 2 1 = 2 5 2 + 2 1 = 2 10 9 .
Rewrite as a single radical: 10 2 9 .
10 2 9
Explanation
Understanding the Problem We are given the expression 5 4 ⋅ 2 and asked to rewrite it as a single radical.
Objective We want to express the given expression as a single radical. To do this, we will first rewrite each radical as a power of 2.
Rewriting the first term First, let's rewrite 5 4 as 4 5 1 . Since 4 = 2 2 , we have ( 2 2 ) 5 1 = 2 5 2 .
Rewriting the second term Next, let's rewrite 2 as 2 2 1 .
Combining the terms Now, the expression is 2 5 2 ⋅ 2 2 1 . Using the property a m ⋅ a n = a m + n , we have 2 5 2 + 2 1 .
Adding the exponents To add the fractions, we need a common denominator. The least common denominator for 5 and 2 is 10. So, we have 5 2 + 2 1 = 10 4 + 10 5 = 10 9 .
Simplifying the exponent Therefore, the expression is now 2 10 9 .
Rewriting as a single radical Finally, we rewrite 2 10 9 as 10 2 9 .
Final Answer Thus, 5 4 ⋅ 2 = 10 2 9 .
Examples
Radicals and exponents are used in various fields such as physics, engineering, and computer science. For example, in physics, the period of a pendulum can be expressed using a square root. In computer graphics, transformations such as scaling and rotations often involve radicals and exponents. Understanding how to manipulate and simplify radical expressions is essential for solving problems in these fields.
