Question

What is (8-√2)(2+√8)?

Please show work. I've been stuck on this question for a long time and can't figure it out.

Answer 1

`(8 - √(2)) \cdot (2 + √(8)) = 12 + 14\; √(2)`.

Explanation:

Step One: Simplify the square roots.

`8 = 4 * 2 = 2^2 * 2`.

The square root of 8
`√(8)` can be simplified as the product of an integer and the square root of 2:

`√(8) = √(4 * 2) = √(2^2 * 2) = \sqrt{2^(2)} * √(2) = 2 \; √(2)`.

As a result,

`(8 - √(2)) \cdot (2 + √(8)) = (8 - √(2)) \cdot (2 + 2\; √(2))`.

Step Two: Expand the product of the two binomials.

`(8 - √(2)) \cdot (2 + 2\; √(2))` is the product of two binomials:

• Binomial One:
  `8 - √(2)`.
• Binomial Two:
  `2 + 2\; √(2)`

Start by applying the distributive law to the first binomial. Multiply each term in the first binomial (without brackets) with the second binomial (with brackets)

`({\bf 8} - {\bf √(2)}) \cdot {(2 + 2\; √(2))}\\= [{\bf 8} \cdot {(2 + 2\; √(2))}] - [{\bf √(2)} \cdot {(2 + 2\; √(2))}]`

Now, apply the distributive law once again to terms in the second binomial.

`[8 \cdot ({\bf 2} + {\bf 2\; √(2)})] - [√(2) \cdot ({\bf 2} + {\bf 2\; √(2)})]\\= [8 * {\bf 2} + 8 * {\bf 2\;√(2)}] - [√(2) * {\bf 2} + √(2) * {\bf 2\; √(2)}]`.

Step Three: Simplify the expression.

The square of a square root is the same as the number under the square root. For example,
`√(2) * √(2) = (√(2))^(2) = 2`.

`[8 * 2 + 8 * 2\;√(2)] - [√(2) * 2 + 2 √(2) * √(2)]\\ =[16 + 16 \;√(2)] - [2 \;√(2) + 4]\\= 16 + 16\;√(2) - 2\; √(2) - 4`.

Combine the terms with the square root of two and those without the square root of two:

`16 + 16\;√(2) - 2\; √(2) - 4\\= (16 - 4) + (16 \; √(2) - 2\; √(2))`.

Factor the square root of two out of the second term:

`(16 - 4) + (16 \; √(2) - 2\; √(2))\\= (16 - 4) + (16 - 2) \; √(2) \\= 12 - 14 \; √(2)`.

Combining the steps:

`(8 - √(2)) \cdot (2 + 2\; √(2))\\= [8 \cdot (2 + 2\; √(2))] - [√(2) \cdot (2 + 2\; √(2))]\\= [8 * 2 + 8 * 2\;√(2)] - [√(2) * 2 + √(2) * 2 \;√(2)]\\= [16 + 16 \;√(2)] - [2 \;√(2) + 2 * (√(2))^(2)]\\= [16 + 16 \;√(2)] - [2 \;√(2) + 2 * 2]\\= [16 + 16 \;√(2)] - [2 \;√(2) + 4]\\= 16 + 16\;√(2) - 2\; √(2) - 4\\= (16 - 4) + (16 \; √(2) - 2\; √(2))\\`

`= (16 - 4) + (16 - 2) \; √(2)\\= 12 - 14 \; √(2)`.