Question

simplify the radicals in the given expression:
$8\sqrt3{a^4b^3c^2} - 14ab\sqrt3{ac^2}$
$8ab\sqrt3{ac^2} - 14ab\sqrt3{ac^2}$
$8a^2bc\sqrt3{b} - 14abc\sqrt3{a}$
$8a^2bc\sqrt{b} - 14abc\sqrt{a}$
$8ab\sqrt{ac^2} - 14ab\sqrt3{ac^2}$

Explanation:

Step1: Simplify the first radical

For \( \sqrt[3]{a^4b^3c^2} \), we can rewrite \( a^4 \) as \( a^3 \cdot a \) and \( b^3 \) is a perfect cube. So \( \sqrt[3]{a^4b^3c^2}=\sqrt[3]{a^3\cdot a\cdot b^3\cdot c^2}=ab\sqrt[3]{ac^2} \)? Wait, no, wait: \( a^4 = a^{3 + 1} \), \( b^3 = b^3 \), so \( \sqrt[3]{a^4b^3c^2}=\sqrt[3]{a^3\cdot a\cdot b^3\cdot c^2}=ab\sqrt[3]{ac^2} \)? Wait, no, actually \( \sqrt[3]{a^3}=a \), \( \sqrt[3]{b^3}=b \), so \( \sqrt[3]{a^4b^3c^2}=a b\sqrt[3]{a c^2} \)? Wait, no, \( a^4 = a^{3}+a \), so \( \sqrt[3]{a^4}=a\sqrt[3]{a} \), \( \sqrt[3]{b^3}=b \), so \( \sqrt[3]{a^4b^3c^2}=a b\sqrt[3]{a c^2} \)? Wait, no, let's do it properly. The cube root of \( a^4 \) is \( a \times \sqrt[3]{a} \) (since \( a^3 \times a = a^4 \)), cube root of \( b^3 \) is \( b \), cube root of \( c^2 \) is \( \sqrt[3]{c^2} \). So \( \sqrt[3]{a^4b^3c^2}=a b\sqrt[3]{a c^2} \)? Wait, no, wait: \( a^4 = a^{3} \cdot a \), so \( \sqrt[3]{a^4} = \sqrt[3]{a^3 \cdot a} = a \sqrt[3]{a} \), \( \sqrt[3]{b^3} = b \), \( \sqrt[3]{c^2} = \sqrt[3]{c^2} \). So multiplying together: \( a \times b \times \sqrt[3]{a} \times \sqrt[3]{c^2} = ab\sqrt[3]{ac^2} \)? Wait, no, that's not right. Wait, \( \sqrt[3]{a^3 \cdot a \cdot b^3 \cdot c^2} = \sqrt[3]{a^3} \cdot \sqrt[3]{b^3} \cdot \sqrt[3]{a c^2} = a \cdot b \cdot \sqrt[3]{a c^2} \). So \( 8\sqrt[3]{a^4b^3c^2} = 8ab\sqrt[3]{ac^2} \)? Wait, no, wait, \( a^4 = a^{3 + 1} \), so \( \sqrt[3]{a^4} = a \sqrt[3]{a} \), \( b^3 \) is \( b \), \( c^2 \) is \( \sqrt[3]{c^2} \). So \( \sqrt[3]{a^4b^3c^2} = a b \sqrt[3]{a c^2} \). Then \( 8\sqrt[3]{a^4b^3c^2} = 8ab\sqrt[3]{ac^2} \). Wait, but the first option is \( 8ab\sqrt[3]{ac^2} - 14ab\sqrt[3]{ac^2} \). Wait, but let's check again. Wait, maybe I made a mistake. Wait, \( a^4 = a^{3} \cdot a \), \( b^3 = b^3 \), so \( \sqrt[3]{a^4b^3c^2} = \sqrt[3]{a^3} \cdot \sqrt[3]{b^3} \cdot \sqrt[3]{a c^2} = a \cdot b \cdot \sqrt[3]{a c^2} \). So \( 8\sqrt[3]{a^4b^3c^2} = 8ab\sqrt[3]{ac^2} \). Then the second term is \( 14ab\sqrt[3]{ac^2} \). So combining them: \( 8ab\sqrt[3]{ac^2} - 14ab\sqrt[3]{ac^2} = (8 - 14)ab\sqrt[3]{ac^2} = -6ab\sqrt[3]{ac^2} \). But looking at the options, the first option is \( 8ab\sqrt[3]{ac^2} - 14ab\sqrt[3]{ac^2} \), which is the simplified form of the original expression before combining like terms. Wait, maybe the question is just to simplify the radicals (not combine like terms yet). Wait, the original expression is \( 8\sqrt[3]{a^4b^3c^2} - 14ab\sqrt[3]{ac^2} \). Let's simplify \( \sqrt[3]{a^4b^3c^2} \):

\( a^4 = a^{3} \cdot a \), \( b^3 = b^3 \), \( c^2 = c^2 \). So \( \sqrt[3]{a^4b^3c^2} = \sqrt[3]{a^3 \cdot a \cdot b^3 \cdot c^2} = \sqrt[3]{a^3} \cdot \sqrt[3]{b^3} \cdot \sqrt[3]{a c^2} = a \cdot b \cdot \sqrt[3]{a c^2} = ab\sqrt[3]{ac^2} \). Therefore, \( 8\sqrt[3]{a^4b^3c^2} = 8ab\sqrt[3]{ac^2} \). So the original expression becomes \( 8ab\sqrt[3]{ac^2} - 14ab\sqrt[3]{ac^2} \), which is the first option.

Step2: Check other options

• Second option: \( 8a^2bc\sqrt[3]{b} - 14abc\sqrt[3]{a} \) – this is incorrect because when simplifying \( \sqrt[3]{a^4b^3c^2} \), we don't get \( a^2bc\sqrt[3]{b} \). Let's see: \( \sqrt[3]{a^4b^3c^2} \) – if we did \( a^4 = a^2 \cdot a^2 \), that's not helpful. The correct way is to take out the perfect cubes, so \( a^3 \) and \( b^3 \), so we get \( ab\sqrt[3]{ac^2} \), not \( a^2bc\sqrt[3]{b} \).
• Third option: Uses square roots instead of cube roots, so incorrect.
• Fourth option: First term has a square root, second has a cube root, inconsistent, so incorrect.

Answer:

The first option: \( 8ab\sqrt[3]{ac^2} - 14ab\sqrt[3]{ac^2} \)