Simplifying Radicals Worksheets - Math Monks - Free Printable

Let’s solve each problem one by one. We’re simplifying square roots of perfect squares — that means we look for numbers and variables that are squared, and take their square root.

Remember:
- √(a²) = a (if a is positive)
- √(xⁿ) = x^(n/2) if n is even
- For fractions: √(a/b) = √a / b
- For products: √(ab) = √a * √b

We’ll go step by step.

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① √(25m²)
√25 = 5, √(m²) = m → 5m

② √(121x¹⁶)
√121 = 11, √(x¹⁶) = x⁸ → 11x⁸

③ √(9x²)
√9 = 3, √(x²) = x → 3x

④ √(64w¹²)
√64 = 8, √(w¹²) = w⁶ → 8w⁶

⑤ -√(100a⁴)
First, √(100a⁴) = 10a², then apply the negative sign → -10a²

⑥ √(ab¹⁰)
√(a⁶) = a³, √(b¹⁰) = b⁵ → a³b⁵

⑦ √(625y⁶⁴)
√625 = 25, √(y⁴) = y³² → 25y³²

⑧ √441
What number squared is 441? 21×21=441 → 21

⑨ √(15a)
15 is not a perfect square, and ‘a’ has no exponent → cannot simplify further → √(15a)

⑩ √(49y⁴m¹⁰⁰)
√49 = 7, √(y) = y², √(m¹⁰⁰) = m⁵⁰ → 7y²m⁵⁰

⑪ (a²b⁴)
√(a²) = a, √(b⁴) = b² → ab²

⑫ (16a²y / m¹⁶)
Break it down:
√16 = 4, √(a²) = a, √(y⁴) = y², √(m¹⁶) = m⁸
So → 4ay² / m⁸

⑬ √(81/36)
√81 = 9, √36 = 6 → 9/6 = 3/2 → 3/2

⑭ √(x⁴ / y²¹⁰)
√(x⁴) = x², √(y²¹⁰) = y¹⁰⁵ → x² / y¹⁰⁵

⑮ (196a⁴)
√196 = 14, √(a) = a² → 14a²

⑯ √(729a¹²b¹⁶)
√729 = 27 (since 27×27=729), √(a¹²)=a⁶, √(b¹⁶)=b → 27a⁶b⁸

⑰ (900/225)
√900 = 30, √225 = 15 → 30/15 = 2 → 2

⑱ √(361a⁴ / 121b⁴)
√361 = 19, √(a⁴) = a², √121 = 11, √(b⁴) = b² → 19a² / 11b²

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Final Answer:
① 5m
② 11x⁸
③ 3x
④ 8w⁶
⑤ -10a²
⑥ a³b⁵
⑦ 25y³²
⑧ 21
⑨ √(15a)
⑩ 7y²m⁵⁰
⑪ ab²
⑫ 4ay²/m⁸
⑬ 3/2
⑭ x²/y¹⁰⁵
⑮ 14a²
⑯ 27a⁶b⁸
⑰ 2
⑱ 19a²/(11b²)