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.

√45 - 3√20 + 4√5 = √[(9)(5)] - 3√[(4)(5)] + 4√5 = 3√5 - (3)(2)√5 + 4√5 = 3√5 - 6√5 + 4√5 = 1√5 = √5.

There is no "shortcut" or simple way of adding those two numbers. The simplest way to indicate that sum is 2 + √3. The only other way to show it is 2 + 1.732, or 3.732.

Let's assume you're asking for this: (√8 + 3) / √2. Whenever there is a radical sign in the denominator, we "rationalize the denominator," meaning we remove the radical sign. In this case we do that by multiplying the numerator and the denominator by √2: (√8 + 3)(√2) / (√2)(√2) = {√[(8)(2)] + 3√2} / √[(2)(2)] = (√16 + 3√2) / √4 = (4 + 3√2) / 2. You can leave the answer in that form, or you can substitute 1.414 for √2, multiply by 3 and add 4 and divide by 2.

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

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!

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.

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

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

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

√x + √x = 2√x.

2√35.

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.

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.

\( \frac{1}{\sqrt{3}} \) can be simplified to \( \frac{\sqrt{3}}{3} \). Then subtract \( \sqrt{3} \) from that to get the final answer.

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.