Q&A for How to Add and Subtract Square Roots

Because (x - y)(x + y) = x² - y², the example you show is equal to 6 - 2, or 4.

√2 + √2 = 2√2 = √(2² x 2) = √8.

√2 + √2 = 2√2.

Here you're subtracting one square-root-of-8 from two square-roots-of-8. That leaves one square-root-of-8. √8 simplifies to 2√2.

2√48 - √3 = [2√(3 x 16)] - (√3) = [2√(3 x 4²)] - (√3) = [(2 x 4)√3] - (√3) = 8√3 - 1√3 = 7√3.

√4 = 2. Add 9.

(√3 + 2√3) ÷ 2 = (3√3) / 2.

Multiply 1.414 by two.

5√2 - 7√2 + 1√2 = -1√2 = -√2.

Change the fractions to equivalent fractions with the same denominator: 23/3 = 46/6, and 3/2 = 9/6. So 23/3 - 3/2 = 46/6 - 9/6 = 57/6.

Add 1.414 plus 1.414.

√3 = 1.732. 1.732 - 1 = 0.732.

2√3 - 1√3 + √12 = 1√3 + √(2² x 3) = 1√3 + 2√3 = 3√3.

It looks like some irrational number, but it turns out to be exactly 10. Here's how. Let x be the value of the expression, and square both sides so that x^2 = (45-√2000) + √ (45+√2000) + 2 √ (45-√2000) √ (45+√2000). Expand the cross term using difference of squares formula and simplify to x^2 = 90 + 2 √(45^2 - 2000) = 90 + 2√25 = 100. Since x is obviously the positive root of 100, x = 10.

2² + √2 = 4 + √2. You can leave the answer in that form, or you can express it as (4 + 1.414) = 5.414.

(6√3) - 6 = (6)(1.732) - 6 = 10.392 - 6 = 4.392.

The object is to eliminate the square root in the denominator. Do this by multiplying the numerator and the denominator by the conjugate of the denominator, which is (4 + 2√3). Thus, [(4 + 2√3) / (4 - 2√3)][(4 + 2√3) / (4 + 2√3)] = (16 + 16√3 + 12) / (16 - 12) = 28 + 16√3) / 4 = 7 + 4√3.

Multiply both numerator and denominator by √3. That's the same as multiplying by 1, which doesn't change the value of the expression. (12 / 7√3)(√3 / √3) = (12√3) / [( 7)(3)] = (12√3 / 21 = (4√3) / 7. There's now a square root (an irrational number) in the numerator but none in the denominator, which is what you want. (A rational denominator is considered easier to work with than an irrational one, although in practice it doesn't really make much difference, since you typically wind up with an irrational number in the numerator. Nonetheless, it is standard practice to rationalize denominators.)

(√5)² + 5² = 5 + 25 = 30.

First rationalize the denominator (get rid of the radical signs there). Do that by multiplying both numerator and denominator by (√5 - √3). The new numerator is (√5 - √3)², and the new denominator is 5 - 3 = 2. You can leave the answer in that form, or you can further evaluate: (2.24 - 1.73)² / 2 = (0.51)² / 2 = 0.26 / 2 = 0.13.

If you start with any number and add it to itself, you wind up with two of those numbers. In other words, x + x = 2x. Therefore, √2 + √2 = 2√2. If you want to evaluate that further, it's (2)(1.414) = 2.828.

That equals (2)(√3) - 10)(√3) = -8√3. You can leave it in that form, or you can further evaluate: (-8)(1.732) = -13.856.

2√35.

√x + √x = 2√x.

Any number - rational or irrational - being subtracted from itself will always equal zero.

These two are "like" terms and can be combined into 9 times the square root of 5x^6.

This can't be simplified, so you will have to use a calculator to find the exact value.

2√6 + 3√54 + 3√24 = 2√6 + 3√[(9)(6)] + 3√[(4)(6)] = 2√6 + (3)(3)√6 + (2)(3)√6 = 2√6 + 9√6 + 6√6 = 17√6. That's a good place to leave it, or you can further evaluate: 17√6 = (17)(2.45) = 41.65.

It doesn't. √2 and √3, cannot be combined (except by evaluating them and adding them together. √2 + √2 = 2√2, simply because when you add something to itself, you get two of those things!

You will have to use a calculator to find the exact answer to this solution. The answer will be 8.683.