Problem 32 - College Algebra with Modeling and Visualization

Question

Exercises \(17-34:\) Evaluate the expression by hand. Check your result with a calculator. $$ \left(\frac{1}{2}\right)^{0} $$

Step-by-Step Solution

Answer

The expression \(\left(\frac{1}{2}\right)^0\) equals 1.

Step 1: Identify the Property

Recognize that any non-zero number raised to the power of zero equals one. This is a fundamental property of exponents.

Step 2: Apply the Exponent Property

Using the property from the previous step, evaluate the expression. Given expression: \(\left(\frac{1}{2}\right)^0\)Using the property that \(a^0 = 1\) for any non-zero \(a\), we have: \[\left(\frac{1}{2}\right)^0 = 1 \]

Step 3: Check with a Calculator

Enter \(\left(\frac{1}{2}\right)^0\) into a calculator to verify the calculation. The result should confirm that the expression evaluates to 1.

Key Concepts

Exponent Rules

Exponent rules are fundamental principles that help us simplify complex expressions involving powers. One of the key rules is that any non-zero number raised to the power of zero equals one. This might seem surprising initially, but it's a crucial part of the exponential functions' behavior.
Let's break this down:

• **Zero Exponent Rule**: For any non-zero number \(a\), \(a^0 = 1\). This holds true because dividing like terms with the same exponent results in subtracting the exponents, and when \(a^n/a^n\) is simplified, it results in \(a^{n-n} = a^0 = 1\).
• **Product of Powers Rule**: When you multiply numbers with the same base, you add the exponents. For example, \(a^m \times a^n = a^{m+n}\).
• **Power of a Power Rule**: When raising a power to another power, multiply the exponents. For example, \((a^m)^n = a^{m \times n}\).

Understanding these rules will make it much easier to evaluate expressions and solve equations involving exponents.

Exponents

Exponents are a way to express repeated multiplication of the same number. For example, \(2^3\) is the same as saying 2 multiplied by itself three times: \(2 \times 2 \times 2\). Recognizing how exponents work is essential for simplifying mathematical expressions.
Here are some tips:

• **Base and Exponent**: The number being multiplied is the 'base' and the number of times it’s multiplied is the 'exponent'. In \(3^4\), 3 is the base and 4 is the exponent.
• **Negative Exponents**: A negative exponent indicates division. For example, \(a^{-n} = 1/a^n\).
• **Fractional Exponents**: These represent roots, \(a^{m/n}\) is the \(n\)-th root of \(a\) raised to the \(m\) power.

Exponents make calculations more efficient, especially with larger numbers, by eliminating repetitive multiplication.

Calculator Verification

Using a calculator to verify your manual calculations is a great way to ensure accuracy. It's especially useful with exponent calculations because it swiftly confirms if complex expressions have been simplified correctly.
Here's how it works:

• **Input Method**: Enter the expression exactly as shown. For instance, for \((\frac{1}{2})^0\), you have to use parentheses correctly to ensure the exponent applies to the whole fraction.
• **Accuracy Check**: After calculating manually, type the expression into your calculator. The outcome should match your manual solution, which for exponent zero, should be 1.
• **Troubleshoot**: If the calculator gives a different result, revisit your steps to check for any miscalculations.

Utilizing a calculator for verification helps build confidence in your math skills and reinforces the concepts you’ve learned.