Question

multiply. assume a, b, and c are greater than or equal to zero, and write your answer in simplest form.
$8\sqrt{7a^{4}b^{4}c^{4}} \cdot \sqrt{66a^{3}b^{5}c^{4}}$

Explanation:

Step1: Multiply the radicands

When multiplying square roots, we can use the property \(\sqrt{m}\cdot\sqrt{n}=\sqrt{m\cdot n}\). So we have:
\(8\sqrt{7a^{4}b^{4}c^{4}}\cdot\sqrt{66a^{3}b^{5}c^{4}} = 8\sqrt{(7a^{4}b^{4}c^{4})\cdot(66a^{3}b^{5}c^{4})}\)
First, multiply the coefficients and the variables separately. For the coefficients: \(7\times66 = 462\). For the variable \(a\): \(a^{4}\cdot a^{3}=a^{4 + 3}=a^{7}\) (using the property \(a^{m}\cdot a^{n}=a^{m + n}\)). For the variable \(b\): \(b^{4}\cdot b^{5}=b^{4+5}=b^{9}\). For the variable \(c\): \(c^{4}\cdot c^{4}=c^{4 + 4}=c^{8}\). So the radicand becomes \(462a^{7}b^{9}c^{8}\), and we have \(8\sqrt{462a^{7}b^{9}c^{8}}\)

Step2: Simplify the square root

Now we simplify \(\sqrt{462a^{7}b^{9}c^{8}}\). We can express each variable with exponents as a product of even and odd exponents (for simplifying square roots, we want to take out the largest perfect square factors).

• For \(a^{7}\): \(a^{7}=a^{6}\cdot a=(a^{3})^{2}\cdot a\)
• For \(b^{9}\): \(b^{9}=b^{8}\cdot b=(b^{4})^{2}\cdot b\)
• For \(c^{8}\): \(c^{8}=(c^{4})^{2}\)

So we can rewrite the radicand as:
\(\sqrt{462\times(a^{3})^{2}\cdot a\times(b^{4})^{2}\cdot b\times(c^{4})^{2}}=\sqrt{462}\times\sqrt{(a^{3})^{2}}\times\sqrt{a}\times\sqrt{(b^{4})^{2}}\times\sqrt{b}\times\sqrt{(c^{4})^{2}}\)
Since \(\sqrt{(x^{n})^{2}}=x^{n}\) for \(x\geq0\) (and we know \(a,b,c\geq0\)), this simplifies to \(a^{3}b^{4}c^{4}\sqrt{462ab}\)

Step3: Multiply by the coefficient outside the square root

Now we multiply this by the 8 outside the square root:
\(8\times a^{3}b^{4}c^{4}\sqrt{462ab}=8a^{3}b^{4}c^{4}\sqrt{462ab}\)
But we can also check if we made a mistake in the coefficient multiplication. Wait, let's re - check the coefficient multiplication. Wait, \(7\times66 = 462\), that's correct. But maybe we can factor 462 further? Let's see, \(462=2\times3\times7\times11\), no perfect square factors. But wait, let's re - examine the exponents again.
Wait, let's re - do the exponent simplification:

• \(a^{7}=a^{6 + 1}=(a^{3})^{2}\cdot a\), correct.
• \(b^{9}=b^{8+1}=(b^{4})^{2}\cdot b\), correct.
• \(c^{8}=(c^{4})^{2}\), correct.

So \(\sqrt{462a^{7}b^{9}c^{8}}=\sqrt{462}\times\sqrt{a^{7}}\times\sqrt{b^{9}}\times\sqrt{c^{8}}=\sqrt{462}\times a^{3}\sqrt{a}\times b^{4}\sqrt{b}\times c^{4}=a^{3}b^{4}c^{4}\sqrt{462ab}\)
Then multiplying by 8 gives \(8a^{3}b^{4}c^{4}\sqrt{462ab}\)

Wait, but let's check the initial multiplication again. Wait, maybe I made a mistake in the coefficient. Wait, \(7\times66 = 462\), that's correct. Alternatively, maybe we can write the final answer as \(8a^{3}b^{4}c^{4}\sqrt{462ab}\) or we can check if we can simplify \(462ab\) further, but since 462 has no square factors, this is the simplified form.

Wait, let's re - check the exponent of \(b\): \(b^{4}\times b^{5}=b^{9}\), and \(\sqrt{b^{9}}=\sqrt{b^{8}\times b}=b^{4}\sqrt{b}\), correct. For \(a^{7}\), \(\sqrt{a^{7}}=\sqrt{a^{6}\times a}=a^{3}\sqrt{a}\), correct. For \(c^{8}\), \(\sqrt{c^{8}}=c^{4}\), correct. And the coefficient 462 is \(7\times66 = 462\), which is correct. So the final expression is \(8a^{3}b^{4}c^{4}\sqrt{462ab}\)

Answer:

\(8a^{3}b^{4}c^{4}\sqrt{462ab}\)