Solve: 3^x × 1/3^(-x) × 3^(2x) Using Laws of Exponents

Solve the following problem:

$3^x\cdot\frac{1}{3^{-x}}\cdot3^{2x}=\text{?}$

First we will perform the multiplication of fractions using the rule for multiplying fractions:

$\frac{a}{b}\cdot\frac{c}{d}=\frac{a\cdot c}{b\cdot d}$

Let's apply this rule to the problem:

$3^x\cdot\frac{1}{3^{-x}}\cdot3^{2x}=\frac{3^x}{1}\cdot\frac{1}{3^{-x}}\cdot\frac{3^{2x}}{1}=\frac{3^x\cdot1\cdot3^{2x}}{1\cdot3^{-x}\cdot1}=\frac{3^x\cdot3^{2x}}{3^{-x}}$

In the first stage we performed the multiplication of fractions and then simplified the resulting expression,

Next let's recall the law of exponents for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

Let's apply this law to the numerator of the expression that we obtained in the last stage:

$\frac{3^x\cdot3^{2x}}{3^{-x}}=\frac{3^{x+2x}}{3^{-x}}=\frac{3^{3x}}{3^{-x}}$

Now let's recall the law of exponents for division between terms with identical bases:

$\frac{a^m}{a^n}=a^{m-n}$

Let's apply this law to the expression that we obtained in the last stage:

$\frac{3^{3x}}{3^{-x}}=3^{3x-(-x)}=3^{3x+x}=3^{4x}$

We applied the above law of exponents carefully, given that the term in the denominator has a negative exponent hence we used parentheses,

Let's summarize the solution so far:

$3^x\cdot\frac{1}{3^{-x}}\cdot3^{2x}=\frac{3^x\cdot3^{2x}}{3^{-x}} = \frac{3^{3x}}{3^{-x}}=3^{4x}$

Recall the law of exponents for power of a power but in the opposite direction:

$a^{m\cdot n}=(a^m)^n$

Let's apply this law to the expression that we obtained in the last stage:

$3^{4x}=3^{4\cdot x}=\big(3^4\big)^x$

When we applied the above law of exponents instead of opening the parentheses and performing the multiplication between the exponents in the exponent (which is the direct way of the above law of exponents), we represented the expression in question as a term with an exponent in parentheses to which an exponent applies.

Therefore the correct answer is answer B.