Final answer:
To distribute and simplify the radicals in the equation, we simplified the square root of 12 and the square root of 8, then multiplied them by 6 and negative square root of 2 respectively, applying multiplication rules for radicals and signs.
Step-by-step explanation:
The question involves distributing and simplifying radicals, which is a topic in mathematics, specifically algebra. To address the question, we need to simplify √12 and √8, and then multiply them by each other and by 6 and √2 respectively, taking into account the rules for multiplication involving radicals and the signs of the numbers.
First, we simplify the radicals:
- √12 can be broken down into √(4×3), which simplifies to 2√3.
- √8 can be broken down into √(4×2), which simplifies to 2√2.
Now we distribute the simplified radicals:
- (2√3 × 6) = 12√3
- (2√3 × -2√2) = -4√6
As per the rules of multiplication:
- When two positive numbers multiply, the answer has a positive sign, e.g., 2x3 = 6.
- When two negative numbers multiply, the answer has a positive sign, e.g., (-4) x (-3) = 12.
- When the two numbers multiplied have opposite signs, the answer has a negative sign, e.g., (-3) x 2 = -6.
Following these rules, we get:
- 12√3 (positive result)
- -4√6 (negative result because the signs are opposite)
It is important to also check the answer to see if it is reasonable, and to eliminate terms wherever possible to simplify the algebra.