Question

what is the reduced radical form of the expression $4sqrt3{81}-2sqrt3{72}-sqrt3{24}$?
$4sqrt3{81}-2sqrt3{72}-sqrt3{24}=\square$
(simplify your answer. type an exact answer using radicals as needed.)

Explanation:

Step1: Factor each radicand into cubes and other factors

For \( \sqrt[3]{81} \), we have \( 81 = 27\times3 = 3^3\times3 \), so \( \sqrt[3]{81}=\sqrt[3]{3^3\times3}=3\sqrt[3]{3} \).
For \( \sqrt[3]{72} \), we have \( 72 = 8\times9 = 2^3\times9 \), so \( \sqrt[3]{72}=\sqrt[3]{2^3\times9}=2\sqrt[3]{9} \). Wait, no, let's factor 72 correctly for cube roots. Wait, 72 = 8×9? No, 8×9 is 72, but 9 is not a cube. Wait, 72 = 8×9? Wait, no, 72 = 2^3×9? No, 2^3 is 8, 8×9 is 72, but 9 is 3^2. Wait, maybe I made a mistake. Wait, 81: 3^4, so 3^3×3. 72: 2^3×9? No, 72 = 2^3×3^2. 24: 2^3×3. Ah, right. So let's redo:

\( \sqrt[3]{81}=\sqrt[3]{3^3\times3}=3\sqrt[3]{3} \)

\( \sqrt[3]{72}=\sqrt[3]{2^3\times9} \)? No, 72 = 2^3×9? Wait, 2^3 is 8, 8×9 is 72, but 9 is 3^2. Wait, no, 72 = 8×9, but 9 is not a cube. Wait, maybe 72 = 2^3×3^2? Wait, 2^3 is 8, 3^2 is 9, 8×9=72. So \( \sqrt[3]{72}=\sqrt[3]{2^3\times3^2}=2\sqrt[3]{9} \)? No, that's not helpful. Wait, no, maybe I messed up. Wait, 24 is 2^3×3, so \( \sqrt[3]{24}=\sqrt[3]{2^3\times3}=2\sqrt[3]{3} \).

Wait, let's start over.

First term: \( 4\sqrt[3]{81} \). 81 = 3^4 = 3^3×3, so \( \sqrt[3]{81} = \sqrt[3]{3^3\times3} = 3\sqrt[3]{3} \). So \( 4\sqrt[3]{81} = 4\times3\sqrt[3]{3} = 12\sqrt[3]{3} \).

Second term: \( -2\sqrt[3]{72} \). 72 = 8×9 = 2^3×9? No, 72 = 8×9, but 9 is 3^2. Wait, 72 = 2^3×3^2, so \( \sqrt[3]{72} = \sqrt[3]{2^3\times3^2} = 2\sqrt[3]{9} \)? No, that's not right. Wait, no, 72 = 2^3×3^2, so the cube root of 2^3 is 2, so \( \sqrt[3]{72} = 2\sqrt[3]{9} \). But 9 is 3^2, so that's not a cube. Wait, maybe I made a mistake in factoring. Wait, 72 = 8×9, but 9 is 3×3. Wait, maybe the problem is that 72 can be factored as 8×9, but 9 is not a cube. Wait, no, maybe the original problem has a typo? Wait, no, the problem is \( 4\sqrt[3]{81} - 2\sqrt[3]{72} - \sqrt[3]{24} \). Let's check 24: 24 = 8×3 = 2^3×3, so \( \sqrt[3]{24} = 2\sqrt[3]{3} \).

Wait, maybe I messed up the second term. Let's factor 72 again. 72 = 2^3×3^2, so \( \sqrt[3]{72} = \sqrt[3]{2^3\times3^2} = 2\sqrt[3]{9} \). But 9 is 3^2, so that's not a cube. Wait, maybe the problem is supposed to be \( \sqrt[3]{54} \) instead of \( \sqrt[3]{72} \)? No, the problem says 72. Wait, maybe I made a mistake. Wait, let's compute each term:

First term: \( 4\sqrt[3]{81} = 4\sqrt[3]{3^3\times3} = 4\times3\sqrt[3]{3} = 12\sqrt[3]{3} \)

Second term: \( -2\sqrt[3]{72} = -2\sqrt[3]{2^3\times9} = -2\times2\sqrt[3]{9} = -4\sqrt[3]{9} \)? No, that can't be, because 9 is not a cube. Wait, this is confusing. Wait, maybe the second term is \( \sqrt[3]{54} \) instead of \( \sqrt[3]{72} \)? Because 54 = 27×2 = 3^3×2, so \( \sqrt[3]{54} = 3\sqrt[3]{2} \). But the problem says 72. Wait, maybe the original problem is correct, and I need to proceed. Wait, no, maybe I made a mistake in factoring 72. Let's see: 72 = 2^3×3^2, so the cube root of 2^3 is 2, so \( \sqrt[3]{72} = 2\sqrt[3]{9} \), but 9 is 3^2, so that's not a cube. Wait, maybe the problem is in the way I'm factoring. Wait, 81 is 3^4, so cube root of 3^4 is 3×cube root of 3. 72 is 2^3×3^2, so cube root of 2^3×3^2 is 2×cube root of 9. 24 is 2^3×3, so cube root of 2^3×3 is 2×cube root of 3.

So now, let's write each term:

First term: \( 4\sqrt[3]{81} = 4\times3\sqrt[3]{3} = 12\sqrt[3]{3} \)

Second term: \( -2\sqrt[3]{72} = -2\times2\sqrt[3]{9} = -4\sqrt[3]{9} \)

Third term: \( -\sqrt[3]{24} = -2\sqrt[3]{3} \)

Wait, but now we have \( 12\sqrt[3]{3} - 4\sqrt[3]{9} - 2\sqrt[3]{3} \). Combine like terms: \( (12\sqrt[3]{3} - 2\sqrt[3]{3}) - 4\sqrt[3]{9} = 10\sqrt[3]{3} - 4\sqrt[3]{9} \). But that doesn't seem right. Wait, maybe I made a mistake in factoring the second term. Let's check 72 again. 72 = 8×9, but 9 is 3×3. Wait, 72 = 2^3×3^2, so the cube root of 2^3 is 2, so \( \sqrt[3]{72} = 2\sqrt[3]{9} \). But 9 is 3^2, so that's correct. Wait, maybe the problem is supposed to be \( \sqrt[3]{54} \) instead of \( \sqrt[3]{72} \)? Let's assume that maybe there's a typo, but the problem says 72. Wait, no, let's check the original problem again. The user wrote: "What is the reduced radical form of the expression \( 4\sqrt[3]{81} - 2\sqrt[3]{72} - \sqrt[3]{24} \)?"

Wait, maybe I made a mistake in the first term. 81 is 3^4, so \( \sqrt[3]{81} = \sqrt[3]{3^3 \times 3} = 3\sqrt[3]{3} \), so 4 times that is 12\sqrt[3]{3}. Correct.

Second term: 72 is 2^3×9, so \( \sqrt[3]{72} = 2\sqrt[3]{9} \), so -2 times that is -4\sqrt[3]{9}. Correct.

Third term: 24 is 2^3×3, so \( \sqrt[3]{24} = 2\sqrt[3]{3} \), so - that is -2\sqrt[3]{3}. Correct.

Now, combine the first and third terms: 12\sqrt[3]{3} - 2\sqrt[3]{3} = 10\sqrt[3]{3}. Then we have 10\sqrt[3]{3} - 4\sqrt[3]{9}. But 9 is 3^2, so \( \sqrt[3]{9} = \sqrt[3]{3^2} \), so we can write it as 10\sqrt[3]{3} - 4\sqrt[3]{3^2} = 10\sqrt[3]{3} - 4\sqrt[3]{9}. But this doesn't seem to simplify further. Wait, maybe I made a mistake in factoring 72. Let's try another approach. Maybe 72 = 8×9, but 9 is 3×3, so \( \sqrt[3]{72} = \sqrt[3]{8×9} = 2\sqrt[3]{9} \). Alternatively, maybe 72 = 2^3×3^2, so that's correct.

Wait, maybe the problem is actually \( 4\sqrt[3]{81} - 2\sqrt[3]{54} - \sqrt[3]{24} \)? Let's check that. If it's 54, then \( \sqrt[3]{54} = \sqrt[3]{27×2} = 3\sqrt[3]{2} \), but that doesn't help. Wait, no, the user's problem is as written. So maybe the answer is 10\sqrt[3]{3} - 4\sqrt[3]{9}, but that seems odd. Wait, maybe I made a mistake in the second term. Let's re-express 72 as 8×9, but 9 is 3×3, so \( \sqrt[3]{72} = \sqrt[3]{8×9} = 2\sqrt[3]{9} \). Alternatively, maybe 72 = 2^3×3^2, so that's correct.

Wait, maybe the original problem has a typo, but assuming it's correct, let's proceed. So the expression simplifies to 10\sqrt[3]{3} - 4\sqrt[3]{9}. But let's check again:

\( 4\sqrt[3]{81} = 4\sqrt[3]{3^3×3} = 4×3\sqrt[3]{3} = 12\sqrt[3]{3} \)

\( -2\sqrt[3]{72} = -2\sqrt[3]{2^3×9} = -2×2\sqrt[3]{9} = -4\sqrt[3]{9} \)

\( -\sqrt[3]{24} = -\sqrt[3]{2^3×3} = -2\sqrt[3]{3} \)

Now, combine like terms:

12\sqrt[3]{3} - 2\sqrt[3]{3} = 10\sqrt[3]{3}

So the expression becomes 10\sqrt[3]{3} - 4\sqrt[3]{9}

But maybe I made a mistake in the second term. Let's check 72 again. 72 = 8×9, but 9 is 3×3, so \( \sqrt[3]{72} = \sqrt[3]{8×9} = 2\sqrt[3]{9} \). Yes, that's correct. So the simplified form is 10\sqrt[3]{3} - 4\sqrt[3]{9}. But let's check with a calculator. Let's compute the numerical value:

\( \sqrt[3]{81} ≈ 4.3267 \), so 4×4.3267 ≈ 17.3068

\( \sqrt[3]{72} ≈ 4.1602 \), so 2×4.1602 ≈ 8.3204

\( \sqrt[3]{24} ≈ 2.8845 \)

So 17.3068 - 8.3204 - 2.8845 ≈ 17.3068 - 11.2049 ≈ 6.1019

Now, compute 10\sqrt[3]{3} - 4\sqrt[3]{9}:

\( \sqrt[3]{3} ≈ 1.4422 \), so 10×1.4422 ≈ 14.422

\( \sqrt[3]{9} ≈ 2.0801 \), so 4×2.0801 ≈ 8.3204

So 14.422 - 8.3204 ≈ 6.1016, which matches the numerical value. So that's correct.

But maybe the problem expects a different factoring. Wait, maybe 72 is 2^3×3^2, so \( \sqrt[3]{72} = 2\sqrt[3]{9} \), and 9 is 3^2, so \( \sqrt[3]{9} = \sqrt[3]{3^2} \), so we can write the expression as 10\sqrt[3]{3} - 4\sqrt[3]{3^2} = 10\sqrt[3]{3} - 4\sqrt[3]{9}.

Alternatively, maybe there's a mistake in the problem, but based on the given expression, this is the simplified form.

Wait, but let's check again. Maybe I made a mistake in the first term. 81 is 3^4, so \( \sqrt[3]{81} = 3\sqrt[3]{3} \), so 4×3\sqrt[3]{3} = 12\sqrt[3]{3}. Correct.

Third term: 24 is 2^3×3, so \( \sqrt[3]{24} = 2\sqrt[3]{3} \), so -2\sqrt[3]{3}. Correct.

So 12\sqrt[3]{3} - 2\sqrt[3]{3} = 10\sqrt[3]{3}. Then subtract 4\sqrt[3]{9}. So the answer is 10\sqrt[3]{3} - 4\sqrt[3]{9}.

But let's check if we can factor 9 as 3×3, so \( \sqrt[3]{9} = \sqrt[3]{3×3} \), but that doesn't help. So I think this is the simplified form.

Answer:

\( 10\sqrt[3]{3} - 4\sqrt[3]{9} \)