Create a mathexproblem of radicals that use multiplication and addition
Simplify the expression:
√(2 + √3) * √(2 - √3) + √(3 + √2) * √(3 - √2)
Step 1: Simplify the first radical expression within the parentheses:
√(2 + √3) = √(2 + √3)
Step 2: Simplify the second radical expression within the parentheses:
√(2 - √3) = √(2 - √3)
Step 3: Simplify the third radical expression within the parentheses:
√(3 + √2) = √(3 + √2)
Step 4: Simplify the fourth radical expression within the parentheses:
√(3 - √2) = √(3 - √2)
Therefore, the simplified expression is:
(√(2 + √3) * √(2 - √3)) + (√(3 + √2) * √(3 - √2))
Does this use multiplication and addition
Apologies for the confusion. You're right; the expression I provided does not explicitly involve addition and multiplication. Let me give you another example that uses both operations:
Simplify the expression:
(√5 + 2) * (√5 - 1) + (√2 + √3) * (√2 - √3)
To simplify, we can expand and combine like terms:
(√5 + 2)(√5 - 1) + (√2 + √3)(√2 - √3)
Using the FOIL method to multiply the binomials:
(√5 * √5) + (√5 * -1) + (2 * √5) + (2 * -1) + (√2 * √2) + (√2 * -√3) + (√3 * √2) + (√3 * -√3)
Simplifying further:
5 - √5 + 2√5 - 2 + 2 - √6 - √6 - 3
Combining like terms:
7√5 - √5 - 6√6 - 1
Consolidating the radicals:
(7-1)√5 - 6√6 - 1
The simplified expression becomes:
6√5 - 6√6 - 1
Now one with division and subtraction
Certainly! Here's an example involving division and subtraction:
Simplify the expression:
(4√10 - √15) / (√5 - √2) - (2√3 - √6)
To simplify, we can use the concept of multiplying by the conjugate to eliminate the square roots in the denominator.
Multiply both the numerator and denominator by the conjugate of the denominator (√5 + √2):
[(4√10 - √15) / (√5 - √2)] * [(√5 + √2) / (√5 + √2)]
Expanding the numerator and denominator:
[(4√10 * √5) + (4√10 * √2) - (√15 * √5) - (√15 * √2)] / [(√5 * √5) - (√5 * √2) + (√2 * √5) - (√2 * √2)]
Simplifying further:
[(4√50) + (4√20) - √75 - √30] / [5 - √10 + √10 - 2]
Combining like terms:
[4√50 + 4√20 - √75 - √30] / [3]
Consolidating the radicals:
[4√2 * √25 + 4√2 * √5 - √5 * √15 - √3 * √10] / [3]
Simplifying further:
[4√2 * 5 + 4√2 * √5 - √5 * √3 - √3 * √10] / [3]
Simplifying the radicals:
[20√2 + 4√10 - √15√3 - √30√3] / [3]
Combining like terms:
[20√2 + 4√10 - √45 - √90] / [3]
Simplifying the radicals:
[20√2 + 4√10 - 3√5 - 3√10] / [3]
Combining like terms:
[20√2 - √10 - 3√5] / [3]
The simplified expression becomes:
(20√2 - √10 - 3√5) / 3