Simplify each expression, using exact forms.
a) [tex]\sqrt{96 w^7}-\sqrt{24 w^7}=[/tex] $\square$
b) [tex]3 \sqrt[4]{405 b^{17}}+7 \sqrt[4]{1280 b^{17}}=[/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 (2)
For part (a), simplify 96 w 7 − 24 w 7 by factoring out perfect squares: 96 w 7 = 4 w 3 6 w and 24 w 7 = 2 w 3 6 w . Combine like terms to get 2 w 3 6 w .
For part (b), simplify 3 4 405 b 17 + 7 4 1280 b 17 by factoring out perfect fourth powers: 3 4 405 b 17 = 9 b 4 4 5 b and 7 4 1280 b 17 = 28 b 4 4 5 b . Combine like terms to get 37 b 4 4 5 b .
The simplified expressions are: a) 2 w 3 6 w b) 37 b 4 4 5 b
Explanation
Problem Analysis We are asked to simplify two expressions involving radicals, assuming all variables represent positive quantities.
Simplifying Part (a) First, let's simplify the expression in part (a): 96 w 7 − 24 w 7 . We need to factor out perfect squares from the radicands.
Factoring the Radicands We can express 96 as 16 ⋅ 6 and 24 as 4 ⋅ 6 . Thus, the expression becomes 16 ⋅ 6 w 6 w − 4 ⋅ 6 w 6 w .
Taking out Perfect Squares Simplifying further, we have 4 w 3 6 w − 2 w 3 6 w .
Combining Like Terms Now, we combine like terms: ( 4 − 2 ) w 3 6 w = 2 w 3 6 w .
Simplifying Part (b) Next, let's simplify the expression in part (b): 3 4 405 b 17 + 7 4 1280 b 17 . We need to factor out perfect fourth powers from the radicands.
Factoring the Radicands We can express 405 as 81 ⋅ 5 and 1280 as 256 ⋅ 5 . Thus, the expression becomes 3 4 81 ⋅ 5 b 16 b + 7 4 256 ⋅ 5 b 16 b .
Taking out Perfect Fourth Powers Simplifying further, we have 3 ⋅ 3 b 4 4 5 b + 7 ⋅ 4 b 4 4 5 b , which simplifies to 9 b 4 4 5 b + 28 b 4 4 5 b .
Combining Like Terms Now, we combine like terms: ( 9 + 28 ) b 4 4 5 b = 37 b 4 4 5 b .
Final Answer Therefore, the simplified expressions are: a) 2 w 3 6 w b) 37 b 4 4 5 b
Examples
Radicals and their simplification are used extensively in physics, particularly in mechanics and electromagnetism. For example, when calculating the kinetic energy of an object or the impedance in an AC circuit, you often encounter expressions involving square roots. Simplifying these expressions allows for easier calculations and a better understanding of the physical quantities involved. This skill is also crucial in fields like engineering, where precise calculations are necessary for designing structures and systems.
The simplified expression for part (a) is 2 w 3 6 w and for part (b) is 37 b 4 4 5 b .
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