Problem 34 - College Algebra with Modeling and Visualization

Question

Exercises \(17-34:\) Evaluate the expression by hand. Check your result with a calculator. $$ \frac{10^{-4}}{4^{-3}} $$

Step-by-Step Solution

Answer

The evaluated expression is \( \frac{4}{625} \).

Step 1: Understand the Expression

We need to evaluate the expression \( \frac{10^{-4}}{4^{-3}} \). This fraction has two parts: the numerator \(10^{-4}\) and the denominator \(4^{-3}\). We will simplify each separately.

Step 2: Simplify the Numerator

The numerator is \(10^{-4}\), which means take 10 to the power of \(-4\). This results in \( \frac{1}{10^4} \) or \( \frac{1}{10000} \).

Step 3: Simplify the Denominator

The denominator is \(4^{-3}\), which means take 4 to the power of \(-3\). This results in \( \frac{1}{4^3} \) or \( \frac{1}{64} \).

Step 4: Set Up the Division

We are dividing the fraction from the numerator by the fraction from the denominator: \( \frac{ \frac{1}{10000} }{ \frac{1}{64} } \). According to division of fractions, this is the same as multiplying the numerator by the reciprocal of the denominator: \( \frac{1}{10000} \times \frac{64}{1} \).

Step 5: Multiply the Fractions

Multiply the numerators and the denominators together: \( \frac{1 \times 64}{10000 \times 1} = \frac{64}{10000} \).

Step 6: Simplify the Resulting Fraction

Simplify \( \frac{64}{10000} \) by finding the greatest common divisor (GCD) of 64 and 10000, which is 16. Dividing both the numerator and the denominator by 16 gives \( \frac{4}{625} \).

Key Concepts

Powers and Exponents

Powers and exponents are fundamental concepts in algebra that allow us to express repeated multiplication. For instance, the expression \(10^{-4}\) involves the base number 10 raised to the power of -4. The negative exponent indicates that the base is being divided (or inverted) rather than multiplied. For this reason, \(10^{-4}\) is equivalent to \(\frac{1}{10^4}\), simplifying to \(\frac{1}{10000}\). Similarly, \(4^{-3}\) means \(\frac{1}{4^3}\), or \(\frac{1}{64}\). Understanding how to manipulate these powers, especially negative exponents, is a key skill in algebra. It allows for the simplification of complex equations and expressions by converting negative exponents into fractions quickly.

• Negative exponents signify taking the reciprocal of the base raised to the positive exponent.
• When calculating \(a^{-n}\), you are essentially working out \(\frac{1}{a^n}\).

Fraction Simplification

Fraction simplification is a technique used to present fractions in their simplest form, ensuring they are easier to understand and work with in calculations. This process involves dividing both the numerator and the denominator by their greatest common divisor (GCD). In the example given, once we computed \(\frac{64}{10000}\) from the division of two fractions, it's important to find the GCD of 64 and 10000, which is 16. By dividing both the numerator and the denominator by 16, we simplify the fraction to \(\frac{4}{625}\). This new fraction is in its simplest form and is the shortest, most comprehensive version that expresses the same quantity.

• Simplification occurs by finding the GCD of both numerator and denominator.
• The goal is to reduce the fraction to its smallest equivalent form, making calculations easier and the expression more concise.

Division of Fractions

When dividing fractions, the procedure becomes straightforward by converting the division into a multiplication problem. This is achieved by multiplying the first fraction by the reciprocal of the second. In our example, we are given the expression \(\frac{\frac{1}{10000}}{\frac{1}{64}}\). Instead of dividing, you simply multiply \(\frac{1}{10000}\) by the reciprocal of \(\frac{1}{64}\), which is \(\frac{64}{1}\). The problem then becomes \(\frac{1}{10000} \times \frac{64}{1}\). This results in multiplying straight across: numerators multiply with numerators and denominators multiply with denominators, leading to \(\frac{64}{10000}\). This approach is very efficient and keeps the process simpler.

• Division involves the "keep, change, flip" rule: keep the first fraction, change the operation from division to multiplication, and flip the second fraction to find its reciprocal.
• Multiplying by the reciprocal simplifies the fraction division into a manageable multiplication problem.