Problem 17 - Precalculus Mathematics for Calculus

Question

Evaluate each expression. (a) \(\left(\frac{5}{3}\right)^{0} 2^{-1}\) (b) \(\frac{2^{-3}}{3^{0}}\) (c) \(\left(\frac{1}{4}\right)^{-2}\)

Step-by-Step Solution

Answer

(a) \(\frac{1}{2}\), (b) \(\frac{1}{8}\), (c) 16.

Step 1: Understanding Exponents

First, remember two important exponent rules: any non-zero number raised to the power of zero is 1, and a negative exponent indicates a reciprocal. For example, \(a^0 = 1\) and \(a^{-b} = \frac{1}{a^b}\).

Step 2: Evaluate Part (a)

In the expression \(\left(\frac{5}{3}\right)^{0} 2^{-1}\), apply the rule of exponents. \(\left(\frac{5}{3}\right)^{0} = 1\), and \(2^{-1} = \frac{1}{2}\). So, we have \(1 \times \frac{1}{2} = \frac{1}{2}\).

Step 3: Evaluate Part (b)

For \(\frac{2^{-3}}{3^{0}}\), calculate each exponent separately. \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\) and \(3^{0} = 1\). Thus, \(\frac{2^{-3}}{3^{0}} = \frac{\frac{1}{8}}{1} = \frac{1}{8}\).

Step 4: Evaluate Part (c)

Convert the expression \(\left(\frac{1}{4}\right)^{-2}\) by using the rule for negative exponents: \(\left(\frac{1}{4}\right)^{-2} = \left(\frac{4}{1}\right)^{2} = 4^2 = 16\).

Key Concepts

Exponent Rules

Exponent rules are fundamental for understanding how to manipulate powers in mathematical expressions. By mastering these rules, you can simplify and solve expressions efficiently. The basic exponent rules are:

• Zero Exponent Rule: Any non-zero number raised to the power of zero equals 1. Thus, for any number \( a \), \( a^0 = 1 \).
• Power of a Power: When raising a power to another power, multiply the exponents. That is, \( (a^m)^n = a^{m \times n} \).
• Product of Powers: When multiplying two exponents with the same base, add the exponents: \( a^m \times a^n = a^{m+n} \).
• Quotient of Powers: When dividing two exponents with the same base, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Negative Exponent: A negative exponent represents the reciprocal of the base raised to the positive exponent: \( a^{-b} = \frac{1}{a^b} \).

Bearing these rules in mind helps enormously in solving exponent-related problems.

Negative Exponents

Negative exponents may seem tricky initially, but they're quite simple once you understand the basic concept. Negative exponents tell you to take the reciprocal of the base to the positive value of the exponent. For example,

• \( a^{-1} = \frac{1}{a} \)
• \( a^{-b} = \frac{1}{a^b} \)

This means when you see a negative exponent in an expression, you simply "flip" the base.
It's crucial to remember that this rule applies regardless of whether the base is a fraction or a whole number. For instance,

• \( 2^{-1} = \frac{1}{2} \)
• \( \left(\frac{1}{4}\right)^{-2} = \left(\frac{4}{1}\right)^2 = 16 \)

Grasping this concept will allow you to confidently tackle expressions involving negative exponents.

Zero Exponent Rule

The zero exponent rule is one of the simplest yet powerful tools in mathematics. It states that any non-zero number raised to the power of zero is equal to 1. Thus,

• \( a^0 = 1 \), where \( a eq 0 \)

This rule is immensely useful for simplifying expressions. Take, for example, the expression \( \left(\frac{5}{3}\right)^0 \). By applying the zero exponent rule, it simplifies directly to 1.
It's important to internalize this rule because it applies universally to all non-zero numbers, fractions included. Understanding why the zero exponent rule works revolves around maintaining consistent mathematical patterns. Consider the pattern:

• \( a^3 = a \times a \times a \)
• \( a^2 = a \times a \)
• \( a^1 = a \)
• \( a^0 = 1 \)

Whenever you confront zeros in exponents, remember this rule to simplify and proceed quickly.