Simplifying Radicals Worksheets - Math Monks - Free Printable

Let’s solve each problem step by step. We’re simplifying square roots of perfect squares — that means we can take the square root of numbers and variables if they are squared (or raised to an even power).

Remember:
- √(a²) = a (if a ≥ 0, which we assume in these problems)
- √(xⁿ) = x^(n/2) if n is even
- For fractions: √(a/b) = √a / b
- For products: √(ab) = √a × b

---

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, √(y⁶⁴) = y³² → 25y³²

⑧ √441
441 = 21² → 21

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

Wait — but let’s double-check: Is 15 a perfect square? No. So yes, this stays as is.

But hold on — maybe I misread? The problem says “Simplifying Radicals: Perfect Square” — so perhaps all should be simplifiable? Let me check again.

Actually, looking at #9: √(15a) — unless there's a typo, it doesn’t simplify. But let’s keep going — maybe later ones confirm pattern.

Actually, wait — perhaps I made a mistake. Let me re-express:

No — 15 is 3×5, neither is a square. So √(15a) is already simplified.

But let’s continue — maybe the worksheet includes some that don’t simplify? Or did I misread?

Looking back at original image description — user said “uploaded an image”, but I’m solving based on text provided. In the list, #9 is written as “√15a” — which likely means √(15a), not (√15)a. Either way, same result.

So we’ll leave it as √(15a). But let’s verify others first.

⑩ √(49y⁴m¹⁰⁰)
√49 = 7, √(y⁴) = y², √(m¹⁰⁰) = m⁵⁰ → 7y²m⁵⁰

⑪ (a²b⁴)
√(a²) = a, √(b⁴) = b² → ab²

⑫ (16a²y / m¹⁶)
Numerator: √(16a²y⁴) = 4ay²
Denominator: √(m¹⁶) = m⁸
→ 4ay² / m⁸

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

⑭ (x⁴ / y|¹⁰) — Wait, what is “y⁴|¹⁰”? That looks like a typo.

Looking back at original input: “⑭ (x⁴ / y|¹⁰)” — probably meant to be y⁴⁰ or y¹⁰? The vertical bar might be a formatting error.

In many worksheets, it’s common to have exponents like y¹⁰. Also, in #14, if it’s y¹⁰, then:

√(x⁴ / y¹⁰) = x² / y⁵

If it were y⁴⁰, then x² / y²⁰ — but y⁴⁰ seems too big.

Wait — looking at other problems: #10 has m¹⁰⁰, #7 has y⁶⁴ — so large exponents are used.

But in #14, it says “y⁴|¹⁰” — possibly a misrendered “y⁴⁰”? Because | might be a separator or font issue.

Alternatively, maybe it’s “y^4 * 10” — but that doesn’t make sense.

Another possibility: in some fonts, superscript 10 might look like |10 — but unlikely.

Wait — let’s compare with #12: denominator was m¹⁶ — written clearly.

Perhaps it’s “y^{40}” — because 4 and 0 together might look like 4|0? But here it’s 4|10.

Actually, looking again: in the user’s message, it’s written as:

“⑭ √(x / y⁴|¹)”

This is ambiguous. But in context of similar problems, and since #10 has m¹⁰⁰, perhaps it’s y⁴⁰? But 4|10 isn't 40.

Wait — another idea: maybe it’s “y^4 times 10” — but that would be unusual.

Or perhaps it’s a typo and should be y¹⁰.

Given that in #11 we have b⁴, #12 m¹⁶, #10 m¹⁰⁰ — exponents vary.

But let’s look at #14 in standard curriculum: often it’s √(x/y¹⁰) = x²/y⁵.

I think the most reasonable assumption is that “y⁴|¹⁰” is a rendering error for “y¹⁰”. Why? Because if it were y⁴⁰, it would be written as y⁴⁰, not y⁴|¹⁰. The | might be a stray character.

Alternatively, in some systems, ^10 might render poorly.

To resolve this, let’s consider the answer format. If we assume it’s y¹⁰, then answer is x²/y⁵.

If it’s y⁴⁰, then x²/y²⁰.

But let’s see problem #18: √(361a⁴ / 121b⁴) — which is clear.

Also, #17: √(900/225) = 30/15 = 2.

Back to #14 — I think it’s safe to assume it’s y¹⁰, as “y⁴|¹” is likely a typo for “y¹⁰”. Perhaps the “4” is part of the fraction line or something.

Wait — another thought: in LaTeX or math notation, sometimes people write y^{4}10 meaning y^{40}, but that’s nonstandard.

Given the ambiguity, but to proceed, I’ll assume it’s y¹⁰, as it’s more common in such worksheets.

So: √(x⁴ / y¹⁰) = x² / y⁵

But let’s note: if it were y⁴⁰, answer would be different. However, since the problem is listed as #14 and others have consistent patterns, and y¹⁰ makes sense, I’ll go with that.

Actually, wait — looking back at the user’s input string:

"⑭ √(x / y⁴|¹)"

The "y⁴|¹⁰" — perhaps it's "y^{40}" and the | is a separator between 4 and 0? Like in some OCR errors.

In many cases, when scanning, "40" might become "4|0", but here it's "4|10" — which is odd.

Another possibility: it's "y^4 * 10", but that would be 10y^4 under the radical, which is messy.

I think the best course is to assume it's a typo and it's meant to be y^{10}, as that fits the pattern of other problems (like #12 has m^{16}, #10 has m^{100}, so exponents are multiples of 2).

Moreover, in #14, if denominator is y^{10}, then sqrt is y^5, which is integer exponent.

Whereas if it's y^{40}, also fine, but less common.

But let's calculate both ways mentally.

If y^{10}: answer x²/y⁵

If y^{40}: answer x²/y²⁰

Now, looking at problem #7: y^{64} -> y^{32}, so large exponents are used.

Problem #10: m^{100} -> m^{50}

So y^{40} is possible.

But "y⁴|¹⁰" — if we interpret | as a digit separator, perhaps it's 410? Unlikely.

Another idea: in some fonts, the exponent 10 might be written with a small 1 and 0, and if the 1 is close to the base, it might look like |.

I recall that in some old printers, superscripts could be misaligned.

Given that, and to match the difficulty level, I think it's intended to be y^{10}. Why? Because if it were y^{40}, it would be written as y^{40}, not y^4|10.

Perhaps the "4" is a mistake, and it's just y^{10}.

Let me check online or standard problems — but since I can't, I'll make a decision.

Notice that in problem #12, it's m^{16}, which is clear.

In #14, if it's y^{10}, then the answer is nice: x²/y⁵.

If it's y^{40}, x²/y^{20}, also nice.

But let's see the next problems.

#15: √(196a⁴) = 14a²

#16: √(729a^{12}b^{16}) — 729 is 27², a^{12} -> a^6, b^{16}->b^8, so 27a^6b^8

#17: √(900/225) = 30/15 = 2

#18: √(361a⁴ / 121b⁴) = 19a² / 11b²

All clean answers.

For #14, if it's y^{10}, answer is x²/y⁵ — clean.

If y^{40}, x²/y^{20} — also clean.

But "y⁴|¹⁰" — perhaps it's "y^{40}" and the | is between 4 and 0, but written as 4|10 by mistake.

Maybe it's "y^4 * 10", but then √(x^4 / (10 y^4)) = x^2 / (y^2 √10) — not nice, and not matching the "perfect square" theme.

So likely, it's y^{40} or y^{10}.

Another clue: in the list, #10 has m^{100}, which is very large, so y^{40} is plausible.

But let's count the characters: "y⁴|¹⁰" — if we remove the |, it's y⁴¹⁰, which is nonsense.

Perhaps it's a typo and it's y^{10}, and the "4" is from the previous number or something.

I think the safest bet is to assume it's y^{10}, as it's a common exponent in such problems.

To confirm, let's see if there's a pattern in the denominators.

In #12, denominator m^{16} — exponent 16.

In #14, if y^{10}, exponent 10.

In #18, b^4.

So mixed exponents.

I'll go with y^{10} for now.

So #14: √(x^4 / y^{10}) = x^2 / y^5

But let's write it as \frac{x^2}{y^5}

Now, continuing.

⑮ √(196a⁴)
√196 = 14, √(a⁴) = a² → 14a²

⑯ √(729a^{12}b^{16})
729 = 27² (since 27*27=729), a^{12} -> a^6, b^{16}->b^8 → 27a^6b^8

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

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

Now, back to #9: √(15a) — still seems unsimplifiable. But let's double-check if 15a could be interpreted differently. Perhaps it's √15 * a, but same thing.

Maybe it's a trick question, or perhaps I missed that 15 is not a square, so it stays.

But in the context, all others simplify nicely, so maybe #9 is √(15a) and it's already simplified.

Perhaps the "a" is meant to be squared? But it's written as "15a", not "15a²".

Another thought: in some notations, if it's √(15a), and a is a variable, it's fine.

So I'll keep it as √(15a).

But let's list all answers and see.

Perhaps for #9, it's √(15) * √a, but that's the same.

I think it's correct.

Now for #14, I need to decide.

Let me try to search my knowledge: in many algebra worksheets, a common problem is √(x^4 / y^{10}) = x^2 / y^5.

Also, y^{10} is (y^5)^2, so perfect square.

Whereas y^{40} is also perfect square, but larger.

Given that "y⁴|¹⁰" might be a misrendering of "y^{10}", I'll assume that.

Perhaps the "4" is a typo and it's "y^{10}".

So I'll go with that.

Final decisions:

① 5m
② 11x⁸
③ 3x
④ 8w⁶
⑤ -10a²
⑥ a³b⁵
⑦ 25y³²
⑧ 21
⑨ √(15a) [cannot simplify]
⑩ 7y²m⁵⁰
⑪ ab²
⑫ 4ay² / m⁸
⑬ 3/2
⑭ x² / y⁵ [assuming y^{10}]
⑮ 14a²
⑯ 27a⁶b⁸
⑰ 2
⑱ 19a² / 11b²

But for #9, is there a possibility that it's √(15) * a, but that's the same as a√15, which is not simpler.

Or perhaps the problem is √(15a^2)? But it's written as "15a", not "15a²".

In the user's input, it's "⑨ √15a" — which in math notation usually means √(15a), not (√15)a.

And since 15a is not a perfect square, it remains.

Similarly, for #14, to be precise, let's note the ambiguity, but for the sake of completing, I'll use y^{10}.

Perhaps the "|" is meant to be a caret or something, but I think it's safe.

Another idea: in some systems, "y^4|10" might mean y to the power of 4 times 10, i.e., y^{40}, as 4*10=40.

That could be! In programming or some contexts, | might denote multiplication, but in math, not really.

But in handwritten notes, sometimes people write exponents concatenated.

Given that, and since 4*10=40, perhaps it's y^{40}.

Then √(x^4 / y^{40}) = x^2 / y^{20}

Which is also a nice answer.

And it matches the scale of other problems like m^{100} in #10.

In #10, m^{100} -> m^{50}, so y^{40} -> y^{20} is consistent.

Whereas y^{10} -> y^5 is smaller.

Also, in #7, y^{64} -> y^{32}, so large exponents are used.

So perhaps "y⁴|¹⁰" means y^{40}, with | separating 4 and 0, but written as 4|10 by error.

Maybe it's "y^{40}" and the "1" is a typo.

I think y^{40} is more likely given the context.

Let me calculate: if y^{40}, then sqrt is y^{20}, so answer x^2 / y^{20}

If y^{10}, x^2 / y^5

Now, looking at problem #12: denominator m^{16} -> m^8, which is half.

Similarly, for consistency, if #14 has y^{40}, then y^{20}.

Also, in the list, #10 has m^{100}, which is very large, so y^{40} is reasonable.

Moreover, "y⁴|¹⁰" — if we ignore the |, it's y410, which is not good, but if we take 4 and 10 as separate, perhaps 40.

I recall that in some OCR software, "40" can be read as "4|0", but here it's "4|10", which might be "40" with an extra 1.

Perhaps it's "y^{10}" and the "4" is from the numerator or something.

To resolve this, let's look at the position: in the user's message, it's "⑭ √(x⁴ / y⁴|¹⁰)" — the "y⁴|¹⁰" is likely "y^{40}" with a formatting error.

I think I'll go with y^{40} because:
- It makes the exponent even and large, like other problems.
- The answer x^2 / y^{20} is clean.
- If it were y^{10}, it might be written as y^{10} without the 4.

So final decision for #14: assume y^{40}, so answer x^2 / y^{20}

Then for #9, still √(15a)

But let's confirm #9: is there any way it simplifies? 15 = 3*5, no squares, a to power 1, so no.

Perhaps the problem is √(15) * a, but that's a√15, which is not simpler than √(15a), and usually we leave it as √(15a) if a is not squared.

In radical simplification, if the radicand has no perfect square factors, it's simplified.

So √(15a) is simplified.

Now, let's list all answers with this adjustment.

Also, for #13: √(81/36) = 9/6 = 3/2, correct.

#17: √(900/225) = 30/15 = 2, correct.

#18: 19a²/(11b²), correct.

So final answers:

① 5m
② 11x⁸
③ 3x
④ 8w⁶
⑤ -10a²
⑥ a³b⁵
⑦ 25y³²
⑧ 21
⑨ √(15a)
⑩ 7y²m⁵⁰
⑪ ab²
⑫ \frac{4ay^2}{m^8}
⑬ \frac{3}{2}
⑭ \frac{x^2}{y^{20}} [assuming y^{40}]
⑮ 14a²
⑯ 27a^6b^8
⑰ 2
⑱ \frac{19a^2}{11b^2}

For #14, to be accurate, since the input is ambiguous, but based on context, I'll use y^{40}.

Perhaps the "|" is a typo and it's "y^{10}", but I think y^{40} is better fit.

Another way: in the sequence, after #12 which has m^{16}, #14 might have y^{40} as a larger exponent.

So I'll stick with that.

Now, for the final answer section, I need to provide all answers.

The user didn't specify how to present, but since it's a list, I'll list them numbered.

Also, for #9, it's already simplified, so we leave it.

Let me write the final answers clearly.

Final Answer:
① \( 5m \)
② \( 11x^8 \)
③ \( 3x \)
④ \( 8w^6 \)
⑤ \( -10a^2 \)
⑥ \( a^3b^5 \)
⑦ \( 25y^{32} \)
⑧ \( 21 \)
⑨ \( \sqrt{15a} \)
⑩ \( 7y^2m^{50} \)
⑪ \( ab^2 \)
⑫ \( \frac{4ay^2}{m^8} \)
⑬ \( \frac{3}{2} \)
⑭ \( \frac{x^2}{y^{20}} \) (assuming the exponent is 40)
⑮ \( 14a^2 \)
⑯ \( 27a^6b^8 \)
⑰ \( 2 \)
⑱ \( \frac{19a^2}{11b^2} \)

For #14, to be precise, since the input was "y⁴|¹⁰", and if we interpret it as y^{40}, then yes.

If the student has the original, they can verify, but based on reasoning, this is best.

Perhaps in some versions, it's y^{10}, but I think for consistency with large exponents, y^{40} is intended.

So I'll go with that.