Solve sqrt(3) × sqrt(12) + 3²: Complete Radical Expression Calculation

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.