Problem 19 - Algebra and Trigonometry

Question

\(7-28\) Evaluate each expression. $$ \left(\frac{5}{3}\right)^{0} 2^{-1} $$

Step-by-Step Solution

Answer

The value of the expression is \( \frac{1}{2} \).

Step 1: Apply Exponent Rules

Recall the rule that any non-zero number raised to the exponent 0 is 1. Therefore, \( \left(\frac{5}{3}\right)^{0} = 1 \).

Step 2: Simplify the Negative Exponent

Use the rule that \( a^{-b} = \frac{1}{a^b} \). Here, \( 2^{-1} = \frac{1}{2} \).

Step 3: Multiply the Results

Now, multiply the results from Step 1 and Step 2: \( 1 \times \frac{1}{2} = \frac{1}{2} \).

Key Concepts

Zero Exponent

The zero exponent rule is one of the foundational concepts in mathematics, especially when working with exponents. Understanding this rule can simplify many calculations. The rule states that any non-zero number raised to the power of zero equals one.
This might look confusing at first, but consider what happens when you continue to divide a number by itself. For example, if you have \[ a^3 = a imes a imes a \]and continue dividing by \(a\), you will eventually reach \(a^{-1}\), and then if you go one step further, you arrive at \(a^0\). The step-by-step reduction confirms that:

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

Therefore, regardless of whether the base is a fraction or a whole number, as long as the base is not zero, raising it to the power of zero will always result in 1.

Negative Exponents

Negative exponents can initially be tricky, but they're quite straightforward once you understand them. The rule for negative exponents is that a number raised to a negative exponent is equal to the reciprocal of that number raised to the opposite positive exponent.For example:\[ a^{-b} = \frac{1}{a^b} \]This means if we take \(2^{-1}\), it converts to \(\frac{1}{2^1}\), which simplifies to \(\frac{1}{2}\).
Think of negative exponents as a convenient way to represent fractions or reciprocal values. This insight is particularly helpful when simplifying expressions because it transforms complex terms into more manageable components.To summarize:

• Negative exponents "flip" the base to its reciprocal form.
• Ensure the base is not zero when applying this rule, as division by zero is undefined.

Multiplication of Fractions

Multiplying fractions follows a simple rule: multiply the numerators together and multiply the denominators together. This straightforward approach ensures you can handle even complex fraction multiplications efficiently.Let's review the steps with an example calculation, such as multiplying the results found in the original exercise: \[ 1 \times \frac{1}{2} = \frac{1 \cdot 1}{1 \cdot 2} = \frac{1}{2} \]Here, the multiplication of numerators gives us \(1\), and the multiplication of denominators results in \(2\). Thus, the product of the fractions \(1\) and \(\frac{1}{2}\) is \(\frac{1}{2}\).
By following this method:

• You can quickly determine the product of any two fractions.
• Always make sure to simplify your final answer if possible by using the greatest common divisor.

Remember, practice makes it much easier to handle complex mathematical operations, including fraction multiplication.