To simplify radicals without a calculator, find the largest perfect square that divides your number, use the rule 𝑎 𝑏 = 𝑎 × 𝑏 𝑎 𝑏 √ = 𝑎 √ × 𝑏 √ to pull the square root of that perfect square out front, and repeat until no more perfect squares remain, often using factor trees or testing perfect squares like 4, 9, 16, 25, etc., as divisors.
How to simplify radicals
Square Root of 75 by Prime Factorization Method
The prime factorisation of 75 is 3×5×5. Thus, √75 = √3. √5. √5 = 5√3.
Find the largest factor in the radicand that is a perfect power of the index. Rewrite the radicand as a product of two factors, using that factor. Use the product rule to rewrite the radical as the product of two radicals. Simplify the root of the perfect power.
Mental Math Square Roots 1. Find the closest square (must be less) to the number you are trying to take the square root of 2. Divide your number by the square root of the closest square (keep the remainder or decimal) 3. Take the average of the quotient you found in step 2 and the square root of the closest square 4.
We can express the square root of 384 in its lowest radical form as 8 √6.
To find a square root, you look for a number that, when multiplied by itself, gives you the original number (e.g., 3 is the square root of 9 because 3 x 3 = 9). You can use a calculator, prime factorization (grouping factors into pairs), or estimation tricks for perfect squares like knowing the last digit of the root (e.g., roots ending in 3 or 7 give squares ending in 9) to find them by hand.
Step by Step Solution: