$\sqrt{75 t^{11} u^3}$ Assume that all variables represent positive real numbers.
Answer (1)
Factor the constant: Rewrite 75 as 5 2 × 3 .
Rewrite variable powers: Express t 11 as t 10 × t and u 3 as u 2 × u .
Take square roots of even powers: Simplify 5 2 t 10 u 2 to 5 t 5 u .
Combine remaining terms: The simplified expression is 5 t 5 u 3 t u .
Explanation
Understanding the Problem We are asked to simplify the expression 75 t 11 u 3 , assuming that all variables represent positive real numbers.
Factoring the Constant First, we can factor the number 75 as 75 = 25 × 3 = 5 2 × 3 . So we can rewrite the expression as 5 2 × 3 t 11 u 3 .
Rewriting the Powers Next, we can rewrite the powers of t and u so that we have even powers. We have t 11 = t 10 × t and u 3 = u 2 × u . Substituting these into the expression, we get 5 2 × 3 × t 10 × t × u 2 × u .
Taking Square Roots Now, we can take the square root of the terms with even powers. We have 5 2 = 5 , t 10 = t 5 , and u 2 = u . So the expression becomes 5 t 5 u 3 t u .
Final Answer Therefore, the simplified expression is 5 t 5 u 3 t u .
Examples
Simplifying radical expressions is useful in various fields, such as physics and engineering, when dealing with quantities involving square roots. For example, when calculating the period of a pendulum, the formula involves a square root. If the length of the pendulum or the acceleration due to gravity are expressed in terms of variables, simplifying the expression can make it easier to analyze and understand the behavior of the pendulum. Also, simplifying radical expressions can be used in geometry when calculating lengths or areas of figures.
