Problem 6 - Precalculus: Functions and Graphs

Question

Express the number in the form \(a / b,\) where \(a\) and \(b\) are integers. $$\left(-\frac{3}{2}\right)^{4}-2^{-4}$$

Step-by-Step Solution

Answer

The expression simplifies to 5.

Step 1: Evaluate \((-\frac{3}{2})^{4}\)

To evaluate the expression \((-\frac{3}{2})^{4}\), multiply \(-\frac{3}{2}\) by itself 4 times: \[(-\frac{3}{2}) \times (-\frac{3}{2}) \times (-\frac{3}{2}) \times (-\frac{3}{2}) = \frac{81}{16}\] The result is positive because a negative number raised to an even power becomes positive. Thus, the value is \(\frac{81}{16}\).

Step 2: Evaluate \(2^{-4}\)

To find \(2^{-4}\), recall that a negative exponent indicates a reciprocal. Thus, \(2^{-4}\) is equal to:\[\frac{1}{{2^{4}}} = \frac{1}{16}\]So, \(2^{-4}\) is \(\frac{1}{16}\).

Step 3: Subtract the Two Fractions

Now subtract \(2^{-4}\) from \((-\frac{3}{2})^{4}\):\[\frac{81}{16} - \frac{1}{16} = \frac{81 - 1}{16} = \frac{80}{16}\] Thus, the expression simplifies to \(\frac{80}{16}\).

Step 4: Simplify the Fraction \(\frac{80}{16}\)

To simplify \(\frac{80}{16}\), divide both the numerator and the denominator by their greatest common divisor (GCD), which is 16.\[\frac{80}{16} = \frac{80 \div 16}{16 \div 16} = \frac{5}{1}\]This simplifies to \(5\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The base is the number being multiplied by itself, and the exponent tells you how many times to multiply the base. When you see something like \((-rac{3}{2})^{4}\), it means \(-\frac{3}{2}\) is multiplied by itself 4 times:

• \((-rac{3}{2}) \times (-\frac{3}{2}) \times (-\frac{3}{2}) \times (-\frac{3}{2})\)

This results in \(\frac{81}{16}\). When the base is negative and the exponent is even, the result is positive because multiplying negatives in pairs gives a positive. Thus, \((-rac{3}{2})^{4}\) simplifies to a positive fraction.

Negative Exponents

Negative exponents indicate that a number should be taken as a reciprocal raised to the opposite positive power. Consider the expression \(2^{-4}\). Here, the \(-4\) exponent instructs us to find the reciprocal of \(2\) raised to the power of \(4\). This is written as:

• \(2^{-4} = \frac{1}{2^{4}}\)

After evaluating \(2^{4} = 16\), we find \(2^{-4} = \frac{1}{16}\). Negative exponents simply flip the fraction, turning large numbers into small fractions, dense with the potential of inverse growth.

Fraction Simplification

Fraction simplification involves rewriting a fraction in its simplest form. To do this, you need to divide the numerator and the denominator by their greatest common factor. In the exercise, after subtracting the two fractions, you get \(\frac{80}{16}\).
Steps to simplify:

• Identify the greatest common divisor (GCD) of \(80\) and \(16\), which is \(16\).
• Divide both the numerator and the denominator by the GCD: \(\frac{80 \div 16}{16 \div 16} = \frac{5}{1}\).

The fraction simplifies to \(5\), which tells us that \(5\) whole parts make up the original fraction, eliminating any fractional component.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. Finding the GCD is essential in simplifying fractions because it helps reduce them to their simplest form. For example, when you simplify \(\frac{80}{16}\), you need the GCD of \(80\) and \(16\).

• List the factors of each number: \(80: 1, 2, 4, 5, 8, 10, 16, 20, 40, 80\) and \(16: 1, 2, 4, 8, 16\).
• The greatest factor common to both lists is \(16\).
• Divide both the numerator and denominator by \(16\) for simplified expression: \(\frac{5}{1}\).

Finding the GCD means you're efficiently reducing fractions, stripping them down to their core simplicity.