Simplify each expression: [tex]$\frac{\sqrt{1323 p^{14} a^{10}}}{\sqrt{7 p^4 a}}=$[/tex]
Answer (1)
Combine the square roots: 7 p 4 a 1323 p 14 a 10
Simplify the fraction: 189 p 10 a 9
Factor 189 and rewrite a 9 : 3 2 ⋅ 21 p 10 a 8 a
Take out the perfect squares: 3 p 5 a 4 21 a
Explanation
Combine Square Roots We are asked to simplify the expression 7 p 4 a 1323 p 14 a 10 . Let's start by combining the square roots into a single square root.
Simplify Fraction Combining the square roots, we get 7 p 4 a 1323 p 14 a 10 . Now, we simplify the fraction inside the square root.
Divide Coefficients and Simplify Variables First, divide the coefficients: 7 1323 = 189 . Next, simplify the variables using the rules of exponents: p 4 p 14 = p 14 − 4 = p 10 and a a 10 = a 10 − 1 = a 9 .
Rewrite Expression So, the expression becomes 189 p 10 a 9 . Now, we need to factor 189 and identify any perfect squares.
Factor 189 We can factor 189 as 189 = 9 ⋅ 21 = 3 2 ⋅ 21 . Thus, we can rewrite the expression as 9 ⋅ 21 p 10 a 9 .
Rewrite a^9 We can also rewrite a 9 as a 8 ⋅ a = ( a 4 ) 2 ⋅ a . So, the expression becomes 3 2 ⋅ 21 p 10 a 8 a .
Take Out Perfect Squares Now, we take out the perfect squares from the square root: 3 p 5 a 4 21 a .
Final Answer Therefore, the simplified expression is 3 p 5 a 4 21 a .
Examples
Simplifying radical expressions is useful in various fields, such as physics and engineering, where complex formulas often need to be simplified for easier calculations. For example, when calculating the period of a pendulum, the formula involves a square root that might need simplification. By simplifying such expressions, engineers and physicists can more easily analyze and understand the behavior of physical systems. This skill is also crucial in computer graphics for optimizing calculations related to lighting and shading models, where square roots are frequently encountered.
