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Math  /  Algebra

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Express the product (7+2)(7−2)(\sqrt{7}+\sqrt{2})(\sqrt{7}-\sqrt{2}) in simplest form.
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Studdy Solution

STEP 1

1. We are given the expression (7+2)(7−2)(\sqrt{7}+\sqrt{2})(\sqrt{7}-\sqrt{2}).
2. We need to simplify this expression using algebraic identities.
3. The expression is in the form of a difference of squares.

STEP 2

1. Recognize and apply the difference of squares formula.
2. Simplify the resulting expression.

STEP 3

Recognize that the expression (7+2)(7−2)(\sqrt{7}+\sqrt{2})(\sqrt{7}-\sqrt{2}) is a difference of squares. The difference of squares formula is:
(a+b)(a−b)=a2−b2 (a+b)(a-b) = a^2 - b^2
In this case, a=7 a = \sqrt{7} and b=2 b = \sqrt{2} .

STEP 4

Apply the difference of squares formula:
(7+2)(7−2)=(7)2−(2)2 (\sqrt{7}+\sqrt{2})(\sqrt{7}-\sqrt{2}) = (\sqrt{7})^2 - (\sqrt{2})^2

STEP 5

Calculate the squares of the terms:
(7)2=7 (\sqrt{7})^2 = 7 (2)2=2 (\sqrt{2})^2 = 2

STEP 6

Subtract the squares:
7−2=5 7 - 2 = 5
The simplest form of the expression is:
5 \boxed{5}

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