Simplify each expression, using exact forms
a) [tex]16 \sqrt{15}-10 \sqrt{15}=[/tex] $\square$
b) [tex]11 \sqrt[3]{n}+2 \sqrt[3]{n}=[/tex] $\square$
c) [tex]11 \sqrt[3]{a}-7 \sqrt[3]{a}=[/tex] $\square$
NOTE: To input [tex]a \sqrt{b}[/tex], type "a sqrt(b)". To input [tex]a \sqrt[3]{c}[/tex], type "a root(b)(c)".
Answer (1)
Combine like terms in each expression by factoring out the radical.
Perform the arithmetic operation (addition or subtraction) on the coefficients.
Write the simplified expression with the new coefficient and the radical.
The simplified expressions are: a) 6 15 , b) 13 3 n , c) 4 3 a .
Explanation
Understanding the Problem We are asked to simplify three expressions involving radicals. The key idea is to combine like terms, treating the radicals as variables.
Simplifying Expression a a) We have 16 15 − 10 15 . We can factor out the 15 to get ( 16 − 10 ) 15 . Then, we perform the subtraction: 16 − 10 = 6 . So, the simplified expression is 6 15 .
Simplifying Expression b b) We have 11 3 n + 2 3 n . We can factor out the 3 n to get ( 11 + 2 ) 3 n . Then, we perform the addition: 11 + 2 = 13 . So, the simplified expression is 13 3 n .
Simplifying Expression c c) We have 11 3 a − 7 3 a . We can factor out the 3 a to get ( 11 − 7 ) 3 a . Then, we perform the subtraction: 11 − 7 = 4 . So, the simplified expression is 4 3 a .
Final Answer Therefore, the simplified expressions are: a) 6 15 b) 13 3 n c) 4 3 a
Examples
Radicals are used in various fields, such as engineering and physics, to represent quantities like the period of a pendulum or the impedance in an electrical circuit. For example, the period T of a simple pendulum is given by T = 2 π g L , where L is the length of the pendulum and g is the acceleration due to gravity. Simplifying expressions with radicals allows engineers and physicists to make calculations and predictions more efficiently.
