Simplify (3 square root of x)(4 cube root of x^2)

Solution:

Step:1

Commence by multiplying the expressions $\left(3 \sqrt{x}\right)$ and $\left(4 \sqrt[3]{x^{2}}\right)$.

Step:1.1

First, multiply the numerical coefficients $3$ and $4$ to get $12\sqrt{x}\sqrt[3]{x^{2}}$.

Step:1.2

Next, we need to express both radicals with a common index, which in this case is $6$.

Step:1.2.1

Convert $\sqrt[3]{x^{2}}$ to exponential form using the rule $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$, resulting in $12x^{\frac{2}{3}}\sqrt{x}$.

Step:1.2.2

Change the exponent of $x^{\frac{2}{3}}$ to have a denominator of $6$, yielding $12x^{\frac{4}{6}}\sqrt{x}$.

Step:1.2.3

Transform $x^{\frac{4}{6}}$ back to radical form as $\sqrt[6]{x^{4}}$, giving us $12\sqrt[6]{x^{4}}\sqrt{x}$.

Step:1.2.4

Rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ using the same rule, resulting in $12\sqrt[6]{x^{4}}x^{\frac{1}{2}}$.

Step:1.2.5

Adjust the exponent of $x^{\frac{1}{2}}$ to match the common index of $6$, obtaining $12\sqrt[6]{x^{4}}x^{\frac{3}{6}}$.

Step:1.2.6

Convert $x^{\frac{3}{6}}$ to radical form as $\sqrt[6]{x^{3}}$, and we have $12\sqrt[6]{x^{4}}\sqrt[6]{x^{3}}$.

Step:1.3

Combine the radicals using the product rule, which gives us $12\sqrt[6]{x^{4}x^{3}}$.

Step:1.4

To simplify further, we multiply $x^{4}$ by $x^{3}$ by adding their exponents.

Step:1.4.1

Apply the power rule $a^{m}a^{n} = a^{m+n}$ to combine the exponents, resulting in $12\sqrt[6]{x^{4+3}}$.

Step:1.4.2

Add the exponents $4$ and $3$ together to get $12\sqrt[6]{x^{7}}$.

Step:2

Factor out $x^{6}$ from under the radical to simplify, which yields $12\sqrt[6]{x^{6}x}$.

Step:3

Finally, extract terms from under the radical according to the properties of radicals, resulting in the simplified expression $12x\sqrt[6]{x}$.

Knowledge Notes:

1. Multiplication of Radicals: When multiplying radicals, you can multiply the coefficients (numbers outside the radical) and the radicands (numbers inside the radical) separately.

2. Least Common Index: When dealing with multiple radicals, it's often necessary to rewrite them with a common index to combine them. The least common index is the smallest index that can be used for all radicals involved.

3. Exponential Form of Radicals: Radicals can be rewritten in exponential form using the rule $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$. This is useful for combining radicals with different indices.

4. Product Rule for Radicals: The product rule states that $\sqrt[n]{a}\sqrt[n]{b} = \sqrt[n]{ab}$, which allows you to multiply radicands while keeping the same index.

5. Power Rule: The power rule for exponents states that $a^{m}a^{n} = a^{m+n}$. This is used to combine like bases with different exponents.

6. Simplifying Radicals: When simplifying radicals, any factors of the radicand that are perfect powers of the index can be taken out from under the radical sign. For example, $\sqrt[6]{x^{6}}$ simplifies to $x$ because $x^{6}$ is a perfect sixth power.