Solve:

$\sqrt{3}\cdot\sqrt{12}+3^2$

**Recall:**

**A. **Defining a root as a power:

$\sqrt[n]{a}=a^{\frac{1}{n}}$Means that all the laws of powers apply to roots as well.

**B.** **Therefore,** we can apply the following rule of powers in which we multiply two different bases with the same exponent:

$x^n\cdot y^n=(x\cdot y)^n$The literal meaning of this law in the given direction is **that we can write a multiplication between two exponents with equal powers as a multiplication between the bases within the exponents raised to the same power**,

**We will apply** these two laws of powers in the problem.

First, **we will convert** all the roots to powers using the definition of a root as a power that was mentioned in **A** above:

$\sqrt{3}\cdot\sqrt{12}+3^2 =3^{\frac{1}{2}}\cdot12^{\frac{1}{2}}+3^2$

Next, we will note that the two exponents in the multiplication have the same power, so we will apply the law of powers mentioned in **B** above:

$3^{\frac{1}{2}}\cdot12^{\frac{1}{2}}+3^2 =(3\cdot12)^\frac{1}{2}+3^2=36^\frac{1}{2} +3^2$

**We will now return** to writing roots using the definition of a root as a power that was mentioned in A above, **but in the opposite direction**:

$a^{\frac{1}{n}} = \sqrt[n]{a}$We will apply this to the expression we obtained:

$36^\frac{1}{2} +3^2 =\sqrt{36}+3^2=6+9=15$For the first term we converted the half power of the first exponent to a square root, for the next we simply calculated (__without a calculator!, that's the whole point here..__) the numerical value of the root.

**In summary**:

$\sqrt{3}\cdot\sqrt{12}+3^2 =3^{\frac{1}{2}}\cdot12^{\frac{1}{2}}+3^2 =36^\frac{1}{2} +3^2=6+9=15$__Therefore, the correct answer is answer C.__