Free Algebra 1 Radical Expressions worksheets for homeschoolers ... - Free Printable

Let’s solve each radical expression step by step. We’ll simplify them by factoring out perfect powers that match the root (like cubes for cube roots, fourths for fourth roots, etc.).

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1. ∛(24z⁴b)

Break down 24:
24 = 8 × 3 → and 8 is a perfect cube (∛8 = 2)
z⁴ = z³ × z → so we can take one z out of the cube root
b stays inside since it has no exponent ≥ 3

So:
∛(24z⁴b) = ∛(8 × 3 × z³ × z × b) = ∛8 × ∛z³ × ∛(3zb) = 2z ∛(3zb) ✔

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2. ⁺√(4802u⁶) — Wait, this is a *fourth* root? The symbol says “⁴√”

Check: 4802 ÷ 2 = 2401 → and 2401 = 7⁴! Because 7×7=49, 49×49=2401 → yes!

So:
4802 = 2 × 2401 = 2 × 7⁴
u⁶ = u⁴ × u² → so we can take u out (since ⁴√u⁴ = u), but u² stays inside

So:
⁴√(4802u⁶) = ⁴√(7⁴ × 2 × u⁴ × u²) = 7u × ⁴√(2u²)

Wait — the answer given in the image is 7u⁴√(2√u) — that doesn’t look right. Let me double-check.

Actually, let’s recompute:

Is 4802 divisible by 16? No. By 81? No. But 7⁴ = 2401 → 2 × 2401 = 4802 → correct.

u⁶ = u⁴ × u² → so ⁴√(u⁶) = u^(6/4) = u^(3/2) = u√u → but we want to write as simplified radical.

Better way:
⁴√(4802u⁶) = ⁴√(7⁴ × 2 × u⁴ × u²) = 7u × ⁴√(2u²)

But ⁴√(2u²) cannot be simplified further unless we write as √(√(2u²)) — which is messy.

Wait — maybe the original problem was meant to be ⁴√(4802u⁸)? Then u⁸ would give u² outside.

But as written: u⁶ → so ⁴√(u⁶) = u^(6/4) = u^(3/2) = u·u^(1/2) = u√u → but under fourth root?

Actually, standard simplification:
We factor out the largest perfect fourth power.

u⁶ = u⁴ · u² → so ⁴√(u⁶) = u · ⁴√(u²)

And ⁴√(u²) = √(√(u²)) = √|u| — but if we assume variables are positive, then √u.

So: ⁴√(4802u⁶) = 7u · ⁴√(2u²) = 7u · √(√(2u²)) — still messy.

Wait — perhaps there's a typo in the problem or answer key? Let’s check numerically.

Try u=1: ⁴√(4802) ≈ ? 7⁴=2401, 8⁴=4096, 9⁴=6561 → so between 8 and 9. ⁴√4802 ≈ 8.3 — not 7.

Wait — I think I made a mistake.

4802 ÷ 2 = 2401 → 2401 = 7^4 → yes.

So ⁴√(4802) = ⁴√(2 × 7^4) = 7 × ⁴√2

Then u⁶: ⁴√(u⁶) = u^(6/4) = u^(3/2) = u√u — but how to write under fourth root?

Actually, ⁴√(u⁶) = ⁴√(u⁴ · u²) = u · ⁴√(u²)

And ⁴√(u²) = (u²)^(1/4) = u^(1/2) = √u — only if we allow fractional exponents.

In radical form, we usually leave it as ⁴√(u²) unless specified.

But the answer given is 7u⁴√(2√u) — which suggests they wrote ⁴√(2u²) as ⁴√(2) · ⁴√(u²) = ⁴√2 · √u — because ⁴√(u²) = √u.

Yes! That’s valid: ⁴√(u²) = (u²)^{1/4} = u^{1/2} = √u

So:
⁴√(4802u⁶) = ⁴√(7⁴ × 2 × u⁴ × u²) = 7u × ⁴√(2) × ⁴√(u²) = 7u × ⁴√2 × √u

But writing √u inside a fourth root? Not standard.

Alternatively, combine: ⁴√(2u²) = ⁴√(2) · ⁴√(u²) = ⁴√2 · √u — so overall: 7u · ⁴√2 · √u = 7u√u · ⁴√2 — still not matching.

Wait — the answer says: 7u⁴√(2√u) — meaning 7u times fourth root of (2 times square root of u)

That would be: 7u · ⁴√(2 · u^{1/2}) = 7u · (2 u^{1/2})^{1/4} = 7u · 2^{1/4} · u^{1/8} — which is not equal to our result.

I think there might be an error in the provided answer key.

Let me recalculate from scratch:

⁴√(4802u⁶)

Factor 4802:
4802 ÷ 2 = 2401
2401 = 7×7×7×7 = 7⁴ → good.

u⁶ = u⁴ × u²

So:
⁴√(7⁴ × 2 × u⁴ × u²) = 7u × ⁴√(2u²)

Now, ⁴√(2u²) = (2u²)^{1/4} = 2^{1/4} u^{1/2} = ⁴√2 · √u

So the simplified form is: 7u · ⁴√2 · √u

But to write as a single radical: 7u · ⁴√(2u²) — which is acceptable.

However, the answer key says: 7u⁴√(2√u) — which implies 7u times fourth root of (2 times sqrt(u)), i.e., 7u · ⁴√(2 u^{1/2})

Which equals 7u · (2 u^{1/2})^{1/4} = 7u · 2^{1/4} u^{1/8} = 7 · 2^{1/4} u^{9/8}

But our expression is 7u · 2^{1/4} u^{1/2} = 7 · 2^{1/4} u^{3/2}

These are not equal — so the answer key likely has a typo.

Perhaps the original problem was ⁴√(4802u⁸)? Then u⁸ gives u² outside, and ⁴√(2) remains, so 7u² ⁴√2 — still not matching.

Or maybe ⁴√(4802u^4) — then 7u ⁴√2 — closer.

Given the confusion, and since the worksheet shows "7u⁴√(2√u)", I suspect they intended to write 7u times the fourth root of (2u²), but miswrote it.

For accuracy, I'll go with mathematically correct simplification.

But let's move on and come back.

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3. ⁴√(1875y⁵)

Factor 1875:
1875 ÷ 5 = 375
375 ÷ 5 = 75
75 ÷ 5 = 15
15 ÷ 5 = 3 → so 1875 = 5⁴ × 3

y⁵ = y⁴ × y

So:
⁴√(5⁴ × 3 × y⁴ × y) = 5y × ⁴√(3y)

Answer given: 5y⁴√(3y) — matches ✔

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4. ∛(1029u)

Factor 1029:
1029 ÷ 3 = 343
343 = 7³ → yes! 7×7=49, 49×7=343

So 1029 = 3 × 7³

Thus:
∛(1029u) = ∛(7³ × 3u) = 7 ∛(3u)

Answer given: 7∛(3u) — matches ✔

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5. ⁴√(4096)

What is 4096?
I know 8⁴ = (8²)² = 64² = 4096 → yes!

So ⁴√(4096) = 8

Answer given: 8 — matches ✔

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6. ⁴√(16a⁴)

16 = 2⁴, a⁴ = a⁴

So ⁴√(2⁴ a⁴) = 2a

Answer given: 2a — matches ✔

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7. ∛(24z⁴o²) — wait, is that "o" or zero? Probably "o" as variable.

Same as first one but with o² instead of b.

24 = 8×3, z⁴ = z³×z, o² stays

So: ∛(8×3×z³×z×o²) = 2z ∛(3zo²)

Answer given: 2z∛(3zo²) — matches ✔

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8. ⁴√(4375e³)

Factor 4375:
4375 ÷ 5 = 875
875 ÷ 5 = 175
175 ÷ 5 = 35
35 ÷ 5 = 7 → so 4375 = 5⁴ × 7

e³ stays as is (less than 4)

So: ⁴√(5⁴ × 7 × e³) = 5 × ⁴√(7e³)

Answer given: 5⁴√(7e³) — matches ✔

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9. ⁴√(4375q⁵)

Same as above: 4375 = 5⁴ × 7

q⁵ = q × q

So: √(5⁴ × 7 × q⁴ × q) = 5q × ⁴√(7q)

Answer given: 5q⁴√(7q) — matches ✔

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10. ⁴√(48z³)

48 = 16 × 3 = 2⁴ × 3

z³ < z⁴, so stays inside

So: ⁴√(2⁴ × 3 × z³) = 2 × ⁴√(3z³)

Answer given: 2⁴√(3z³) — matches ✔

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Now back to #2: ⁴√(4802u⁶)

As calculated: 4802 = 2 × 7⁴, u⁶ = u⁴ × u²

So: ⁴√(7⁴ × 2 × u⁴ × u²) = 7u × ⁴√(2u²)

Now, ⁴√(2u²) = (2u²)^{1/4} = 2^{1/4} u^{1/2} = ⁴√2 · √u

So the expression is 7u · ⁴√2 · √u

To write as a single radical: 7u · ⁴√(2u²)

But the answer key says: 7u⁴√(2√u) — which would be 7u times fourth root of (2 times square root of u)

That is: 7u · ⁴√(2 · u^{1/2}) = 7u · (2 u^{1/2})^{1/4} = 7u · 2^{1/4} u^{1/8} = 7 · 2^{1/4} u^{9/8}

Whereas our correct simplification is 7u · 2^{1/4} u^{1/2} = 7 · 2^{1/4} u^{3/2}

Since u^{3/2} ≠ u^{9/8}, the answer key is incorrect for this one.

Perhaps it's a typo, and it should be ⁴√(4802u⁸)? Then u⁸ gives u² outside, and ⁴√(2) remains, so 7u² ⁴√2 — still not matching.

Or maybe ⁴√(4802u^4) — then 7u ⁴√2 — close to what they have if they forgot the u² part.

Another possibility: they meant to write 7u times the fourth root of (2u²), but wrote "2√u" by mistake.

Because ⁴√(2u²) = ⁴√2 · ⁴√(u²) = ⁴√2 · √u — so if they wrote ⁴√(2√u), that's different.

I think for correctness, we should put the mathematically accurate answer.

But since the worksheet provides answers, and most match, perhaps for this one, we accept their version with note.

Wait — let's calculate numerical value.

Assume u=16 (so that sqrt(u)=4, easy)

Then ⁴√(4802 * 16^6) = ⁴√(4802 * (2^4)^6) = ⁴√(4802 * 2^{24})

But better: u=16, u⁶ = (2^4)^6 = 2^{24}

4802 = 2 * 7^4

So total inside: 2 * 7^4 * 2^{24} = 7^4 * 2^{25}

⁴√(7^4 * 2^{25}) = 7 * 2^{25/4} = 7 * 2^{6 + 1/4} = 7 * 64 * 2^{1/4} = 448 * ⁴√2

Now, their answer: 7u ⁴√(2√u) = 7*16 * ⁴√(2 * √16) = 112 * ⁴√(2*4) = 112 * ⁴√8

⁴√8 = 8^{1/4} = (2^3)^{1/4} = 2^{3/4}

So 112 * 2^{3/4}

But earlier we have 448 * 2^{1/4} = 448 / 2^{3/4} * 2^{1} wait no.

448 * 2^{1/4} vs 112 * 2^{3/4}

Compute ratio: (448 * 2^{1/4}) / (112 * 2^{3/4}) = 4 * 2^{-1/2} = 4 / √2 = 2√2 ≈ 2*1.414=2.828 ≠ 1

So not equal. Therefore, the answer key is wrong for #2.

Correct answer should be 7u ⁴√(2u²) or equivalently 7u \sqrt[4]{2u^2}

But to match format, perhaps write as 7u times fourth root of (2u squared).

Since the other answers are given, and this is the only discrepancy, I'll provide the correct mathematical answer.

For the purpose of this response, since the user might be checking against the worksheet, I'll note the discrepancy but give the correct simplification.

However, looking back at the image, the answer is written as "7u⁴√(2√u)" — which might be a typesetting error, and they meant "7u \sqrt[4]{2u^2}" but wrote "2\sqrt{u}" by mistake.

In many fonts, u² and √u might look similar if handwritten, but here it's typed.

Another thought: perhaps "2√u" means 2 times square root of u, and they intend ⁴√(2 * u^{1/2}), but as shown, it's incorrect.

I think for accuracy, I'll use the correct simplification.

But let's see the instruction: "Solve the problem accurately."

So I must give the correct answer.

Final decision for #2: 7u \sqrt[4]{2u^2}

But to write in the same style as others, perhaps "7u⁴√(2u²)"

In the answer key, they have "7u⁴√(2√u)" — which is different.

Perhaps in the context, they consider ⁴√(u²) = √u, so ⁴√(2u²) = ⁴√2 * √u, and they write it as ⁴√(2√u) — but that's not standard notation.

⁴√(2√u) means fourth root of (2 times square root of u), which is (2 u^{1/2})^{1/4} = 2^{1/4} u^{1/8}, while we need 2^{1/4} u^{1/2}.

So it's off by a factor of u^{3/8}.

Therefore, I conclude the answer key has an error for this item.

For the final answer, I'll list all correct simplifications.

Also, in #7, it's "o" which might be confused with zero, but in context, it's a variable.

Now, compiling all:

1. ∛(24z⁴b) = 2z ∛(3zb)

2. ⁴√(4802u⁶) = 7u ⁴√(2u²) [corrected]

3. ⁴√(1875y⁵) = 5y ⁴√(3y)

4. ∛(1029u) = 7 ∛(3u)

5. ⁴√(4096) = 8

6. ⁴√(16a⁴) = 2a

7. ∛(24z⁴o²) = 2z ∛(3zo²) [assuming o is variable]

8. ⁴√(4375e³) = 5 ⁴√(7e³)

9. ⁴√(4375q⁵) = 5q ⁴√(7q)

10. ⁴√(48z³) = 2 ⁴√(3z³)

For #2, since the worksheet says "7u⁴√(2√u)", but it's incorrect, I'll provide the correct one.

Perhaps in some contexts, they simplify ⁴√(u²) as √u, and write the whole thing as 7u * ⁴√2 * √u, but that's not a single radical.

The instruction is to simplify, and typically we leave it as 7u ⁴√(2u²).

I think for consistency, I'll box the answers as per correct math.

But to match the format, let's write them as in the worksheet but corrected for #2.

Final Answer:

1. \( 2z \sqrt[3]{3zb} \)

2. \( 7u \sqrt[4]{2u^2} \) (note: worksheet has typo)

3. \( 5y \sqrt[4]{3y} \)

4. \( 7 \sqrt[3]{3u} \)

5. \( 8 \)

6. \( 2a \)

7. \( 2z \sqrt[3]{3zo^2} \)

8. \( 5 \sqrt[4]{7e^3} \)

9. \( 5q \sqrt[4]{7q} \)

10. \( 2 \sqrt[4]{3z^3} \)

For the response, since the user might expect the worksheet answers, but the instruction is to solve accurately, I'll provide correct ones.

In the final answer section, I'll list them as per calculation.

Also, for #2, to make it clear, but since the format should be concise, I'll just put the correct expression.

Perhaps the "2√u" in the worksheet is meant to be "2u^2" but miswritten.

I think it's safe to go with mathematically correct.

So for Final Answer, I'll list all ten.

But the user didn't specify which one; the image has multiple problems.

The task is to solve the problems in the image.

So I need to provide all answers.

In the response, I can list them.

Since the ExplainAnswer is false, I don't need to explain, but the critical rule is to work through step by step before final answer, which I did.

Now for Final Answer section, I'll provide the list.

To save space, perhaps write in order.

Final Answer:
1. \( 2z \sqrt[3]{3zb} \)
2. \( 7u \sqrt[4]{2u^2} \)
3. \( 5y \sqrt[4]{3y} \)
4. \( 7 \sqrt[3]{3u} \)
5. \( 8 \)
6. \( 2a \)
7. \( 2z \sqrt[3]{3zo^2} \)
8. \( 5 \sqrt[4]{7e^3} \)
9. \( 5q \sqrt[4]{7q} \)
10. \( 2 \sqrt[4]{3z^3} \)

For #2, I'm confident this is correct.

If the worksheet has a different answer, it's likely a typo.

So I'll go with that.