Simplify each expression, using exact forms
a) [tex]$\sqrt{6 y}-13 \sqrt{6 y}+3 \sqrt{6 y}=[/tex] $\square$
b) [tex]$12 \sqrt[3]{11 b n}+18 \sqrt[3]{11 b n}-13 \sqrt[3]{11 b n}=[/tex] $\square$
NOTE: To input [tex]$a \sqrt{b}$[/tex], type "a sqrt(b)". To input [tex]$\alpha \sqrt[3]{c}$[/tex], type "a root(b)(c)".
Answer (2)
The problem requires simplifying expressions with radicals by combining like terms.
Combine the coefficients of 6 y : 1 − 13 + 3 = − 9 , resulting in − 9 6 y .
Combine the coefficients of 3 11 bn : 12 + 18 − 13 = 17 , resulting in 17 3 11 bn .
The simplified expressions are − 9 6 y and 17 3 11 bn .
Explanation
Understanding the Problem We are asked to simplify two expressions involving radicals. The first expression involves the square root, and the second expression involves the cube root. We will combine like terms in each expression by adding and subtracting the coefficients of the radicals.
Simplifying the First Expression For the first expression, we have 6 y − 13 6 y + 3 6 y . We combine the coefficients: 1 − 13 + 3 = − 9 . Therefore, the simplified expression is − 9 6 y .
Simplifying the Second Expression For the second expression, we have 12 3 11 bn + 18 3 11 bn − 13 3 11 bn . We combine the coefficients: 12 + 18 − 13 = 17 . Therefore, the simplified expression is 17 3 11 bn .
Final Answer Therefore, the simplified expressions are: a) − 9 6 y b) 17 3 11 bn
Examples
Radicals are used in various fields such as engineering, physics, and computer graphics. For example, when calculating the impedance of an AC circuit, you often encounter expressions involving square roots. Simplifying these expressions helps in easier computation and analysis of the circuit's behavior. Similarly, in computer graphics, radicals are used in calculating distances and lighting effects, where efficient simplification can significantly improve performance.
The simplified expression for a) is − 9 6 y and for b) is 17 3 11 bn .
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