Problem 45 - Elementary and Intermediate Algebra

Question

Simplify. Do not use negative exponents in the answer. \(\frac{2^{-4}}{3^{-1}}\)

Step-by-Step Solution

Answer

The simplified form is \( \frac{3}{16} \).

Step 1: Change Negative Exponents to Positive

To simplify the expression, we need to eliminate the negative exponents. Recall the rule for negative exponents: \[ a^{-n} = \frac{1}{a^n} \]Apply this rule to both terms in the fraction:\[ \frac{2^{-4}}{3^{-1}} = \frac{1}{2^4} \div \frac{1}{3^1} \]

Step 2: Perform Division of Fractions

Dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, convert the division into multiplication:\[ \frac{1}{2^4} \div \frac{1}{3} = \frac{1}{2^4} \times 3 \]

Step 3: Simplify the Expression

Now perform the multiplication:\[ \frac{1}{2^4} \times 3 = \frac{3}{2^4} \]Calculate the power of 2 in the denominator:\[ 2^4 = 16 \]This leads to the simplified expression:\[ \frac{3}{16} \]

Key Concepts

Negative Exponents

Understanding negative exponents is crucial when simplifying algebraic expressions. A negative exponent signifies that the base should be reciprocated—flipped over to form a fraction. This is based on the exponent rule:

• \( a^{-n} = \frac{1}{a^n} \)

For example, \(2^{-4}\) is equivalent to \(\frac{1}{2^4}\). Similarly, \(3^{-1}\) turns into \(\frac{1}{3}\).
Rather than perform operations with negative exponents directly, it's always easier to convert them into positive exponents by moving the base to the reciprocal. This adjustment makes subsequent calculations more straightforward and eliminates potential confusion that comes with handling negative powers.

Dividing Fractions

Dividing fractions might seem troublesome at first, but it can be simplified considerably using a straightforward approach. When you need to divide fractions, you can alternatively perform a multiplication :

• \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \)

This operation involves flipping the second fraction (also called taking its reciprocal) and then multiplying as you usually would.
In our problem, the division \(\frac{1}{2^4} \div \frac{1}{3}\) is converted into \(\frac{1}{2^4} \times 3\). This switch simplifies the expression greatly and is a fundamental technique in simplifying algebraic fractions.

Multiplying by a Reciprocal

Multiplying by a reciprocal is a key operation in manipulating algebraic expressions, particularly in solving division problems involving fractions. The reciprocal of a number, or fraction, is obtained by flipping its numerator and denominator.
For example, the reciprocal of \(\frac{1}{3}\) is \(3\) because it's simply the number \(1\) divided by \(3\). Instead of dividing by \(\frac{1}{3}\), multiply by \(3\) to get the same result, in this case, \(\frac{1}{2^4} \times 3\), which simplifies down the fraction significantly.

• Reciprocal of \(x\) is \(\frac{1}{x}\)
• Reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\)

This technique is particularly beneficial because it turns the division into a simpler multiplication problem, thereby streamlining the calculations.