Model the entire radical $\sqrt{80}$ as a mixed radical. Record the value of the coefficient.
Answer (1)
∙ Find the prime factorization of 80, which is 2 4 × 5 .
∙ Rewrite the square root as 80 = 2 4 × 5 .
∙ Simplify the square root: 2 4 × 5 = 4 5 .
∙ The coefficient of the mixed radical is 4 .
Explanation
Understanding the Problem We are asked to express the radical 80 as a mixed radical and find the value of the coefficient. A mixed radical is a radical expression written in the form a b , where a is a rational number and b is an integer.
Prime Factorization To express 80 as a mixed radical, we need to find the prime factorization of 80. The prime factorization of 80 is 2 4 × 5 .
Rewriting the Square Root Now we can rewrite the square root as 80 = 2 4 × 5 .
Simplifying the Square Root We can simplify the square root by taking out the perfect square factors: 2 4 × 5 = 2 4 × 5 = 2 2 × 5 = 4 5 .
Finding the Coefficient The mixed radical form of 80 is 4 5 . The coefficient of the mixed radical is 4.
Final Answer Therefore, the value of the coefficient is 4 .
Examples
Mixed radicals are useful in simplifying expressions and estimating values. For example, suppose you need to find the length of the diagonal of a rectangle with sides of length 20 and 45 . You can simplify these radicals to 2 5 and 3 5 , respectively. Then, using the Pythagorean theorem, the length of the diagonal is ( 2 5 ) 2 + ( 3 5 ) 2 = 20 + 45 = 65 . This simplification makes it easier to work with the numbers and estimate the value.
