Question

Write an expression for the product (6–√)(15−−√)

without a perfect square factor in the radicand.
A. 35–√
B. 310−−√
C. 95–√
D. 910

Answer 1

To find the product of √6 and √15 without a perfect square factor in the radicand, we multiply them to get √90, then simplify to 3√10 after removing the perfect square factor, which is choice B.

Step-by-step explanation:

We are asked to write an expression for the product of two square roots √6 and √15 without a perfect square factor in the radicand. To do this, we can simply multiply the two square roots together. According to the properties of square roots, the product of two square roots is the square root of the product of the two numbers. Therefore, the expression is √(6×15), which simplifies to √90. However, 90 has a perfect square factor of 9 (9×10 = 90). We can further simplify this to √9 × √10, which is 3√10, since √9 is equal to 3.

To ensure there are no perfect square factors within the radicand, we check that 10 does not have any perfect square factors other than 1. Since it doesn't, the expression 3√10 is our final answer, which corresponds to option B.

Answer 2

Answer: The expression (6-√)(15-√) can be simplified by using the distributive property of multiplication over addition:

(6-√)(15-√) = 6 * 15 + 6 * (-√) + (-√) * 15 + (-√) * (-√)

Expanding the first two terms:

6 * 15 = 90

6 * (-√) = -6√

Next, we'll use the identity for the product of two square roots:

(-√) * 15 + (-√) * (-√) = 15√ - √^2

Since the square root of a positive number is positive, √^2 = √. So:

15√ - √^2 = 15√ - √

Putting everything together:

(6-√)(15-√) = 90 - 6√ + 15√ - √

Combining like terms:

(6-√)(15-√) = 90 + 9√

So, the expression (6-√)(15-√) without a perfect square factor in the radicand is:

(6-√)(15-√) = 90 + 9√

Therefore, the answer is:

(6-√)(15-√) = A. 35 + 9√.

Step-by-step explanation: