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

Question

Solve the following problem:

3x13x32x=? 3^x\cdot\frac{1}{3^{-x}}\cdot3^{2x}=\text{?}

Video Solution

Solution Steps

00:00 Simplify the following problem
00:05 In order to remove the negative exponent
00:10 We'll invert the numerator and the denominator and the exponent will become positive
00:13 We'll apply this formula to our exercise
00:18 When multiplying powers with equal bases
00:21 The exponent of the result equals the sum of the exponents
00:24 We'll apply this formula to our exercise, we'll then proceed to add up the exponents
00:38 When there's a power raised to a power, the combined exponent is the product of the exponents
00:41 We'll apply this formula to our exercise
00:45 This is the solution

Step-by-Step Solution

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

abcd=acbd \frac{a}{b}\cdot\frac{c}{d}=\frac{a\cdot c}{b\cdot d}

Let's apply this rule to the problem:

3x13x32x=3x113x32x1=3x132x13x1=3x32x3x 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:

aman=am+n 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:

3x32x3x=3x+2x3x=33x3x \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:

aman=amn \frac{a^m}{a^n}=a^{m-n}

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

33x3x=33x(x)=33x+x=34x \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:

3x13x32x=3x32x3x=33x3x=34x 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:

amn=(am)n a^{m\cdot n}=(a^m)^n

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

34x=34x=(34)x 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.

Answer

(34)x (3^4)^x