Summary
Highlights
Even if radicals initially appear different, they can sometimes be simplified to create like terms. For example, 3√8 - 5√18 can be simplified by breaking down 8 into 4x2 and 18 into 9x2, resulting in 6√2 - 15√2, which simplifies to -9√2.
Learn how to add and subtract radical expressions. Radicals can only be combined if they have the same radical part (e.g., 4√5 + 6√5 = 10√5). If the radical parts are different, they cannot be directly added or subtracted, similar to how 4x + 6y cannot be combined unless they possess the same variable.
Practice a more complex problem: 4√12 + 3√27 - 2√48. Simplify each radical by finding perfect square factors (12=4x3, 27=9x3, 48=16x3). This reveals a common radical, √3, allowing you to combine them to 9√3.
The principles extend to cube roots. When given an expression like 8³√16 + 5³√54 - 3³√128, break down the numbers inside the cube roots into perfect cube factors (16=8x2, 54=27x2, 128=64x2). This leads to a common cube root of 2, allowing simplification to 19³√2.
When multiplying a radical by an expression, distribute. For example, √3 * (7 + √3) becomes 7√3 + √9, which simplifies to 7√3 + 3. Another example, 4√5 * (√7 - √3), results in 4√35 - 4√15.
Multiplying conjugate radical expressions (e.g., (3 + √5)(3 - √5)) simplifies nicely because the middle terms cancel out. This leaves you with the product of the first terms and the product of the last terms, simplifying to 9 - 5 = 4.
When squaring a binomial like (2 + √3)², expand it to (2 + √3)(2 + √3). Since these are not conjugates, the middle terms will not cancel. FOIL (First, Outer, Inner, Last) completely to get 4 + 2√3 + 2√3 + 3, which simplifies to 7 + 4√3.
For expressions like (4√3 + 2)³, expand it as (4√3 + 2)(4√3 + 2)(4√3 + 2). First, multiply the initial two terms. Then, multiply the resulting trinomial by the third binomial, performing multiple distributions and combining like terms to get the final simplified answer of 296 + 240√3.