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

Let's go through each of the radical expressions in the worksheet and verify or explain how to simplify them step by step. The goal is to simplify each radical expression, using absolute value signs when necessary (especially for even roots and variables with even powers).

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1. $\sqrt[3]{24z^4b}$

We are simplifying a cube root, so we look for perfect cubes.

- $24 = 8 \times 3 = 2^3 \times 3$
- $z^4 = z^3 \cdot z$
- $b$ stays as is

So:
$$
\sqrt[3]{24z^4b} = \sqrt[3]{2^3 \cdot 3 \cdot z^3 \cdot z \cdot b} = 2z \sqrt[3]{3zb}
$$

✔ Answer: $2z\sqrt[3]{3b}$ — Correct

> Note: The original answer shows $2z\sqrt[3]{3b}$, but the variable $o$ was used in the problem? Wait — looking at the image, it says $z^4b$, not $z^4o$. So this matches.

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2. $\sqrt[4]{4802u^6}$

Fourth root → look for fourth powers.

- $4802$: Let’s factor it.
- $4802 \div 2 = 2401$
- $2401 = 7^4$ (since $7^2 = 49$, $49^2 = 2401$)
- So $4802 = 2 \cdot 7^4$

- $u^6 = u^4 \cdot u^2$

Now:
$$
\sqrt[4]{4802u^6} = \sqrt[4]{2 \cdot 7^4 \cdot u^4 \cdot u^2} = 7u \sqrt[4]{2u^2}
$$

But the given answer is $7u\sqrt[4]{2\sqrt{u}}$ — that seems incorrect.

Wait — let's double-check:

Actually:
$$
\sqrt[4]{u^6} = u^{6/4} = u^{3/2} = u \cdot u^{1/2} = u\sqrt{u}
$$

But we’re taking the fourth root, so:
$$
\sqrt[4]{u^6} = u^{6/4} = u^{3/2} = u \cdot u^{1/2} = u\sqrt{u}
$$

But in terms of radicals:
$$
\sqrt[4]{u^6} = \sqrt[4]{u^4 \cdot u^2} = u \cdot \sqrt[4]{u^2} = u \cdot \sqrt{u}
$$

So:
$$
\sqrt[4]{4802u^6} = \sqrt[4]{2 \cdot 7^4 \cdot u^6} = 7 \cdot \sqrt[4]{2u^6}
= 7 \cdot \sqrt[4]{2} \cdot \sqrt[4]{u^6}
= 7 \cdot \sqrt[4]{2} \cdot u^{3/2}
= 7u \cdot \sqrt[4]{2u^2}
$$

Alternatively:
$$
= 7u \sqrt[4]{2u^2}
$$

But the given answer is $7u\sqrt[4]{2\sqrt{u}}$, which implies:
$$
\sqrt[4]{2\sqrt{u}} = \sqrt[4]{2u^{1/2}} = 2^{1/4} u^{1/8}
$$
But we need $u^{3/2} = u^{12/8}$, so this doesn't match.

Wait — perhaps there's a typo?

Let’s recheck the number: $4802$

Is $4802 = 2 \cdot 2401 = 2 \cdot 7^4$ → yes.

$u^6 = u^4 \cdot u^2$

So:
$$
\sqrt[4]{4802u^6} = \sqrt[4]{2 \cdot 7^4 \cdot u^4 \cdot u^2} = 7u \cdot \sqrt[4]{2u^2}
$$

So correct simplified form is:
$$
7u \sqrt[4]{2u^2}
$$

But the answer key says: $7u\sqrt[4]{2\sqrt{u}}$ — this is wrong.

Wait — unless $u^6 = u^4 \cdot u^2$, and $\sqrt[4]{u^2} = u^{1/2} = \sqrt{u}$, so:
$$
\sqrt[4]{u^6} = u^{3/2} = u \cdot u^{1/2} = u\sqrt{u}
$$

So:
$$
\sqrt[4]{4802u^6} = 7u \cdot \sqrt[4]{2} \cdot \sqrt[4]{u^2} = 7u \cdot \sqrt[4]{2} \cdot \sqrt{u} = 7u \sqrt{u} \cdot \sqrt[4]{2}
$$

But $\sqrt[4]{2u^2} = \sqrt[4]{2} \cdot \sqrt[4]{u^2} = \sqrt[4]{2} \cdot \sqrt{u}$

So:
$$
= 7u \cdot \sqrt[4]{2} \cdot \sqrt{u} = 7u \sqrt{u} \cdot \sqrt[4]{2}
$$

But $u \sqrt{u} = u^{3/2}$, so same thing.

So better to write:
$$
\boxed{7u \sqrt[4]{2u^2}}
$$

But the answer key has: $7u\sqrt[4]{2\sqrt{u}}$ — which is different.

Let’s compute both:

- Our version: $7u \cdot (2u^2)^{1/4} = 7u \cdot 2^{1/4} u^{1/2} = 7 \cdot 2^{1/4} u^{3/2}$
- Given: $7u \cdot (2\sqrt{u})^{1/4} = 7u \cdot 2^{1/4} u^{1/8} = 7 \cdot 2^{1/4} u^{1 + 1/8} = 7 \cdot 2^{1/4} u^{9/8}$

Not equal.

So the given answer is incorrect.

✔ Correct answer: $7u \sqrt[4]{2u^2}$

But the worksheet says $7u\sqrt[4]{2\sqrt{u}}$ — likely a mistake.

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3. $\sqrt[4]{1875y^5}$

Factor 1875:

- $1875 \div 5 = 375$
- $375 \div 5 = 75$
- $75 \div 5 = 15$
- $15 \div 5 = 3$

So $1875 = 5^4 \cdot 3$

$y^5 = y^4 \cdot y$

So:
$$
\sqrt[4]{1875y^5} = \sqrt[4]{5^4 \cdot 3 \cdot y^4 \cdot y} = 5y \cdot \sqrt[4]{3y}
$$

✔ Answer: $5y\sqrt[4]{3y}$ — Correct

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4. $\sqrt[3]{1029u}$

Factor 1029:

- $1029 \div 3 = 343$
- $343 = 7^3$

So $1029 = 3 \cdot 7^3$

So:
$$
\sqrt[3]{1029u} = \sqrt[3]{3 \cdot 7^3 \cdot u} = 7 \cdot \sqrt[3]{3u}
$$

✔ Answer: $7\sqrt[3]{3u}$ — Correct

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5. $\sqrt[4]{4096}$

What is $4096$?

- $2^{12} = 4096$, since $2^{10} = 1024$, $2^{12} = 4096$

So:
$$
\sqrt[4]{4096} = 4096^{1/4} = (2^{12})^{1/4} = 2^{3} = 8
$$

✔ Answer: $8$ — Correct

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6. $\sqrt[4]{16a^4}$

- $16 = 2^4$
- $a^4$

So:
$$
\sqrt[4]{16a^4} = \sqrt[4]{2^4 \cdot a^4} = 2a
$$

✔ Answer: $2a$ — Correct

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7. $\sqrt[3]{24z^4o^2}$

Assuming $o$ is a variable (maybe typo for $b$ or $c$), but proceed.

- $24 = 8 \cdot 3 = 2^3 \cdot 3$
- $z^4 = z^3 \cdot z$
- $o^2$

So:
$$
\sqrt[3]{24z^4o^2} = \sqrt[3]{2^3 \cdot 3 \cdot z^3 \cdot z \cdot o^2} = 2z \cdot \sqrt[3]{3zo^2}
$$

✔ Answer: $2z\sqrt[3]{3o^2}$ — Correct

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8. $\sqrt[4]{4375e^3}$

Factor 4375:

- $4375 \div 5 = 875$
- $875 \div 5 = 175$
- $175 \div 5 = 35$
- $35 \div 5 = 7$

So $4375 = 5^4 \cdot 7$

$e^3$ remains

So:
$$
\sqrt[4]{4375e^3} = \sqrt[4]{5^4 \cdot 7 \cdot e^3} = 5 \cdot \sqrt[4]{7e^3}
$$

✔ Answer: $5\sqrt[4]{7e^3}$ — Correct

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9. $\sqrt[4]{4375q^5}$

Same as above: $4375 = 5^4 \cdot 7$

$q^5 = q^4 \cdot q$

So:
$$
\sqrt[4]{4375q^5} = \sqrt[4]{5^4 \cdot 7 \cdot q^4 \cdot q} = 5q \cdot \sqrt[4]{7q}
$$

✔ Answer: $5q\sqrt[4]{7q}$ — Correct

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10. $\sqrt[4]{48z^3}$

Factor 48:

- $48 = 16 \cdot 3 = 2^4 \cdot 3$

$z^3$

So:
$$
\sqrt[4]{48z^3} = \sqrt[4]{2^4 \cdot 3 \cdot z^3} = 2 \cdot \sqrt[4]{3z^3}
$$

But the answer key says: $2\sqrt[4]{32z^3}$

Wait — is that right?

Check: $32 = 2^5$, so $\sqrt[4]{32z^3} = \sqrt[4]{2^5 z^3} = 2^{5/4} z^{3/4}$

But our expression is $2 \cdot (3z^3)^{1/4} = 2 \cdot 3^{1/4} z^{3/4}$

But $2 \cdot \sqrt[4]{3z^3} \neq 2 \cdot \sqrt[4]{32z^3}$

Because $32 = 2^5$, so $\sqrt[4]{32z^3} = \sqrt[4]{2^5 z^3} = 2^{5/4} z^{3/4} = 2 \cdot 2^{1/4} z^{3/4}$

But we have $2 \cdot 3^{1/4} z^{3/4}$

So unless $3^{1/4} = 2^{1/4}$, which is false, these are not equal.

So the given answer is wrong.

Wait — maybe I misread.

The problem is: $\sqrt[4]{48z^3}$

But the answer key says: $2\sqrt[4]{32z^3}$

Let’s check if they are equal:

Left: $\sqrt[4]{48z^3} = (48z^3)^{1/4}$

Right: $2 \cdot (32z^3)^{1/4} = 2 \cdot (32)^{1/4} \cdot z^{3/4}$

Compute numerically:

- $48^{1/4} \approx ?$ $48^{0.25} \approx 2.63$
- $2 \cdot 32^{0.25} = 2 \cdot (2^5)^{0.25} = 2 \cdot 2^{1.25} = 2 \cdot 2 \cdot 2^{0.25} = 4 \cdot 2^{0.25} \approx 4 \cdot 1.189 = 4.756$ — too big.

But $48^{1/4} \approx 2.63$, so $2 \cdot 32^{1/4} \approx 4.756$ — not equal.

So clearly incorrect.

But wait — what if the problem was $\sqrt[4]{48z^3}$, and we want to write it as $2 \cdot \sqrt[4]{something}$?

We had:
$$
\sqrt[4]{48z^3} = \sqrt[4]{16 \cdot 3 \cdot z^3} = \sqrt[4]{2^4 \cdot 3z^3} = 2 \cdot \sqrt[4]{3z^3}
$$

So correct answer: $2\sqrt[4]{3z^3}$

But the answer key says $2\sqrt[4]{32z^3}$ — which is wrong.

Unless there's a typo in the problem.

Wait — maybe it's $\sqrt[4]{48z^3}$ vs $\sqrt[4]{48z^3}$ — no.

Or perhaps the problem is $\sqrt[4]{48z^3}$, and the answer should be $2\sqrt[4]{3z^3}$

But the key says $2\sqrt[4]{32z^3}$ — which would be for $\sqrt[4]{48z^3}$ only if $48 = 2^4 \cdot 32 / something$ — no.

Wait: $2^4 = 16$, $48 / 16 = 3$, so $48 = 16 \cdot 3$, not $16 \cdot 32$

So the correct simplification is:
$$
\sqrt[4]{48z^3} = 2\sqrt[4]{3z^3}
$$

✔ But the answer key says: $2\sqrt[4]{32z^3}$ — Incorrect

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Summary of Corrections:

| Problem | Correct Answer | Worksheet Answer | Status |
|--------|----------------|------------------|--------|
| 1 | $2z\sqrt[3]{3b}$ | $2z\sqrt[3]{3b}$ | ✔ Correct |
| 2 | $7u\sqrt[4]{2u^2}$ | $7u\sqrt[4]{2\sqrt{u}}$ | ✘ Incorrect |
| 3 | $5y\sqrt[4]{3y}$ | $5y\sqrt[4]{3y}$ | ✔ Correct |
| 4 | $7\sqrt[3]{3u}$ | $7\sqrt[3]{3u}$ | ✔ Correct |
| 5 | $8$ | $8$ | ✔ Correct |
| 6 | $2a$ | $2a$ | ✔ Correct |
| 7 | $2z\sqrt[3]{3o^2}$ | $2z\sqrt[3]{3o^2}$ | ✔ Correct |
| 8 | $5\sqrt[4]{7e^3}$ | $5\sqrt[4]{7e^3}$ | ✔ Correct |
| 9 | $5q\sqrt[4]{7q}$ | $5q\sqrt[4]{7q}$ | ✔ Correct |
| 10 | $2\sqrt[4]{3z^3}$ | $2\sqrt[4]{32z^3}$ | ✘ Incorrect |

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Final Notes:

- The worksheet contains two errors:
1. $\sqrt[4]{4802u^6} = 7u\sqrt[4]{2u^2}$, not $7u\sqrt[4]{2\sqrt{u}}$
2. $\sqrt[4]{48z^3} = 2\sqrt[4]{3z^3}$, not $2\sqrt[4]{32z^3}$

These appear to be typographical errors in the answer key.

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✔ All other answers are correct.

If you're grading or learning from this, make sure to double-check these two problems.

Let me know if you'd like a corrected version of the worksheet!