Simplify each expression, using exact forms.
a) $(8 \sqrt{5})(8 \sqrt{25})= \square$
b) $(-5 \sqrt[3]{25})(-2 \sqrt[3]{15})= \square$
NOTE: To input $a \sqrt{b}$, type "a sqrt(b)". To input $\alpha \sqrt[3]{c}$, type "a root(b"
Answer (2)
Simplify ( 8 5 ) ( 8 25 ) by simplifying 25 to 5, then multiplying the constants and the radical: 8 × 8 × 5 × 5 = 320 5 .
Simplify ( − 5 3 25 ) ( − 2 3 15 ) by multiplying the constants to get 10, then multiplying the radicals: 3 25 × 3 15 = 3 375 = 5 3 3 , and finally combining them: 10 × 5 3 3 = 50 3 3 .
The simplified form of ( 8 5 ) ( 8 25 ) is 320 5 .
The simplified form of ( − 5 3 25 ) ( − 2 3 15 ) is 50 3 3 .
Explanation
Problem Analysis We are asked to simplify two expressions involving radicals. We will simplify each expression separately, showing all steps.
Simplifying Expression a a) We need to simplify ( 8 5 ) ( 8 25 ) .
First, we simplify 25 which is equal to 5. Then, we have ( 8 5 ) ( 8 × 5 ) .
Now, we multiply the constants: 8 × 8 × 5 = 64 × 5 = 320 .
Finally, we multiply by the radical: 320 5 .
Simplifying Expression b b) We need to simplify ( − 5 3 25 ) ( − 2 3 15 ) .
First, we multiply the constants: ( − 5 ) × ( − 2 ) = 10 .
Then, we multiply the radicals: 3 25 × 3 15 = 3 25 × 15 = 3 5 2 × 3 × 5 = 3 5 3 × 3 = 5 3 3 .
Finally, we combine the constants and the radicals: 10 × 5 3 3 = 50 3 3 .
Final Answer Therefore, the simplified expressions are: a) 320 5 b) 50 3 3
Examples
Radicals are used in various fields such as engineering, physics, and computer graphics. For example, when calculating the distance between two points in a 2D plane, the formula involves square roots. Simplifying radical expressions helps in obtaining exact and simplified forms of these distances, which is crucial in applications like mapping and spatial analysis. Also, in physics, many formulas involve radicals, such as the period of a pendulum or the calculation of energy levels in quantum mechanics. Simplifying these expressions allows for easier computation and better understanding of the physical phenomena.
The simplified form of ( 8 5 ) ( 8 25 ) is 320 5 and the simplified form of ( − 5 3 25 ) ( − 2 3 15 ) is 50 3 3 .
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