Question

simplify:
\\(16w^{3}\sqrt{w} + 19\sqrt{w^{11}}\\)
assume that the variable represents a positive real number.

Explanation:

Step1: Simplify \(\sqrt{w^{11}}\)

Using the property of radicals \(\sqrt{a^m}=a^{\frac{m}{2}}\) (for \(a>0\)), we can rewrite \(w^{11}\) as \(w^{10}\cdot w\). Then \(\sqrt{w^{11}}=\sqrt{w^{10}\cdot w}=\sqrt{w^{10}}\cdot\sqrt{w}\). Since \(\sqrt{w^{10}} = w^5\) (because \((w^5)^2=w^{10}\) and \(w>0\)), we have \(\sqrt{w^{11}} = w^5\sqrt{w}\).

Step2: Substitute back into the original expression

The original expression is \(16w^3\sqrt{w}+19\sqrt{w^{11}}\). Substituting \(\sqrt{w^{11}} = w^5\sqrt{w}\) into it, we get \(16w^3\sqrt{w}+19w^5\sqrt{w}\).

Step3: Factor out the common term

Both terms have a common factor of \(w^3\sqrt{w}\)? Wait, no, wait. Wait, let's check again. Wait, \(16w^3\sqrt{w}\) and \(19w^5\sqrt{w}\) have a common factor of \(w^3\sqrt{w}\)? Wait, no, \(w^5 = w^{3 + 2}\), so \(19w^5\sqrt{w}=19w^{3+2}\sqrt{w}=19w^3\cdot w^2\sqrt{w}\). Wait, actually, the common factor is \(w^3\sqrt{w}\)? Wait, no, let's see: \(16w^3\sqrt{w}+19w^5\sqrt{w}=w^3\sqrt{w}(16 + 19w^2)\)? Wait, no, wait, \(w^5 = w^{3+2}\), so \(19w^5\sqrt{w}=19w^3\cdot w^2\sqrt{w}\), and \(16w^3\sqrt{w}=16w^3\sqrt{w}\). So factoring out \(w^3\sqrt{w}\), we get \(w^3\sqrt{w}(16 + 19w^2)\)? Wait, no, that's not right. Wait, wait, no, let's do it again. Wait, \(\sqrt{w^{11}}=w^5\sqrt{w}\), so the second term is \(19w^5\sqrt{w}\). Now, the first term is \(16w^3\sqrt{w}\), the second is \(19w^5\sqrt{w}\). So we can factor out \(w^3\sqrt{w}\) from both terms? Wait, \(16w^3\sqrt{w}=w^3\sqrt{w}\times16\), and \(19w^5\sqrt{w}=w^3\sqrt{w}\times19w^2\). So factoring out \(w^3\sqrt{w}\), we have \(w^3\sqrt{w}(16 + 19w^2)\)? Wait, no, that seems off. Wait, maybe I made a mistake in simplifying \(\sqrt{w^{11}}\). Let's re - express \(w^{11}\) as \(w^{10 + 1}\), so \(\sqrt{w^{11}}=\sqrt{w^{10}\cdot w}=\sqrt{w^{10}}\cdot\sqrt{w}=w^5\sqrt{w}\) (since \(w>0\), \(\sqrt{w^{10}}=(w^{10})^{\frac{1}{2}}=w^5\)). So the second term is \(19w^5\sqrt{w}\). Now, the first term is \(16w^3\sqrt{w}\), the second is \(19w^5\sqrt{w}\). Now, we can factor out \(w^3\sqrt{w}\) from both terms:

\(16w^3\sqrt{w}+19w^5\sqrt{w}=w^3\sqrt{w}(16 + 19w^2)\)? Wait, no, \(19w^5\sqrt{w}=19w^{3 + 2}\sqrt{w}=19w^3\cdot w^2\sqrt{w}\), and \(16w^3\sqrt{w}=16w^3\sqrt{w}\times1\). So factoring out \(w^3\sqrt{w}\), we get \(w^3\sqrt{w}(16 + 19w^2)\)? Wait, but let's check the exponents again. Wait, \(w^3\times w^2 = w^{3 + 2}=w^5\), which matches the exponent in the second term. And \(w^3\times1 = w^3\), which matches the exponent in the first term. So that's correct. Wait, but maybe there's a better way. Wait, alternatively, maybe I made a mistake in the initial simplification. Wait, let's start over.

Original expression: \(16w^3\sqrt{w}+19\sqrt{w^{11}}\)

First, simplify \(\sqrt{w^{11}}\):

Using the property \(\sqrt{a^n}=a^{\frac{n}{2}}\) for \(a>0\), \(n = 11\), so \(\sqrt{w^{11}}=w^{\frac{11}{2}}=w^{5+\frac{1}{2}}=w^5\cdot w^{\frac{1}{2}}=w^5\sqrt{w}\) (since \(w^{\frac{1}{2}}=\sqrt{w}\)).

So the expression becomes \(16w^3\sqrt{w}+19w^5\sqrt{w}\).

Now, factor out the greatest common factor (GCF) of the two terms. The GCF of \(16w^3\sqrt{w}\) and \(19w^5\sqrt{w}\) is \(w^3\sqrt{w}\) (since the GCF of the coefficients 16 and 19 is 1, and the GCF of \(w^3\) and \(w^5\) is \(w^3\), and the GCF of \(\sqrt{w}\) and \(\sqrt{w}\) is \(\sqrt{w}\)).

So factoring out \(w^3\sqrt{w}\), we get:

\(w^3\sqrt{w}(16 + 19w^2)\)

Wait, but let's check if that's correct. Let's distribute back:

\(w^3\sqrt{w}\times16=16w^3\sqrt{w}\)

\(w^3\sqrt{w}\times19w^2 = 19w^{3 + 2}\sqrt{w}=19w^5\sqrt{w}\)

Which matches the original expression after simplifying \(\sqrt{w^{11}}\). So that's correct.

Wait, but maybe there's a miscalculation here. Wait, let's check the exponents again. The first term is \(16w^3\sqrt{w}=16w^{3+\frac{1}{2}}=16w^{\frac{7}{2}}\)

The second term, after simplifying \(\sqrt{w^{11}}\), is \(19w^5\sqrt{w}=19w^{5+\frac{1}{2}}=19w^{\frac{11}{2}}\)

Now, let's express both terms with the same exponent. Let's see, \(\frac{7}{2}\) and \(\frac{11}{2}\). Wait, \(w^{\frac{11}{2}}=w^{\frac{7}{2}+2}=w^{\frac{7}{2}}\cdot w^2\)

So \(19w^{\frac{11}{2}}=19w^2\cdot w^{\frac{7}{2}}\)

And \(16w^{\frac{7}{2}}\) is just \(16w^{\frac{7}{2}}\)

So now, factoring out \(w^{\frac{7}{2}}\), we get:

\(w^{\frac{7}{2}}(16 + 19w^2)\)

But \(w^{\frac{7}{2}}=w^{3+\frac{1}{2}}=w^3\sqrt{w}\), which is the same as before. So that's correct.

So the simplified form is \(w^3\sqrt{w}(16 + 19w^2)\) or \( (16 + 19w^2)w^3\sqrt{w}\), which can also be written as \(w^3\sqrt{w}(19w^2 + 16)\) (order of terms in the parentheses doesn't matter).

Wait, but let's check with a specific value of \(w\). Let's take \(w = 2\).

Original expression: \(16\times2^3\sqrt{2}+19\sqrt{2^{11}}\)

Calculate \(2^3 = 8\), so \(16\times8\sqrt{2}=128\sqrt{2}\)

\(2^{11}=2048\), \(\sqrt{2048}=\sqrt{1024\times2}=32\sqrt{2}\), so \(19\times32\sqrt{2}=608\sqrt{2}\)

Sum: \(128\sqrt{2}+608\sqrt{2}=736\sqrt{2}\)

Now, using the simplified expression: \(w^3\sqrt{w}(16 + 19w^2)\) with \(w = 2\)

\(2^3\sqrt{2}(16 + 19\times2^2)=8\sqrt{2}(16 + 19\times4)=8\sqrt{2}(16 + 76)=8\sqrt{2}\times92 = 736\sqrt{2}\)

Which matches. So the simplification is correct.

Answer:

\(\boldsymbol{w^3\sqrt{w}(16 + 19w^2)}\) (or equivalently \(\boldsymbol{(16 + 19w^2)w^3\sqrt{w}}\))