Math Problem Statement

Simplify (3+ root3) (2+root2)

Solution

To simplify the expression (3+3)(2+2)(3 + \sqrt{3})(2 + \sqrt{2}), we need to apply the distributive property (also known as the FOIL method for binomials).

Step-by-step:

(3+3)(2+2)=3(2)+3(2)+3(2)+3(2)(3 + \sqrt{3})(2 + \sqrt{2}) = 3(2) + 3(\sqrt{2}) + \sqrt{3}(2) + \sqrt{3}(\sqrt{2})

Now, let's compute each term:

  1. 3(2)=63(2) = 6
  2. 3(2)=323(\sqrt{2}) = 3\sqrt{2}
  3. 3(2)=23\sqrt{3}(2) = 2\sqrt{3}
  4. 3(2)=3×2=6\sqrt{3}(\sqrt{2}) = \sqrt{3 \times 2} = \sqrt{6}

Thus, combining all the terms:

6+32+23+66 + 3\sqrt{2} + 2\sqrt{3} + \sqrt{6}

Final Answer:

6+32+23+66 + 3\sqrt{2} + 2\sqrt{3} + \sqrt{6}

Would you like further details on the steps or have any other questions? Here are some related questions:

  1. How do you simplify binomials with irrational numbers in general?
  2. What is the significance of the distributive property in algebra?
  3. How can you apply FOIL in solving other algebraic expressions?
  4. What are the properties of square roots that were used here?
  5. How do you simplify expressions involving square roots in general?

Tip: When multiplying binomials with square roots, always remember to simplify the square root products whenever possible!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Simplification of expressions
Distributive Property
Square Roots
Binomial Expansion (FOIL method)

Formulas

(a + b)(c + d) = ac + ad + bc + bd

Theorems

Distributive property
Properties of square roots

Suitable Grade Level

Grades 7-9