Problem 56 - Intermediate Algebra

Question

Use your calculator to evaluate each numerical expression. $$ 2^{10} $$

Step-by-Step Solution

Answer

The result of \(2^{10}\) is 1024.

Step 1: Understand the Problem

The problem asks us to evaluate the numerical expression \(2^{10}\). This is an exponentiation problem where 2 is the base and 10 is the exponent.

Step 2: Apply Exponent Rules

In this step, recognize that exponentiation indicates that the base number, 2, should be multiplied by itself 10 times. This means we need to calculate \(2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\).

Step 3: Use Calculator

Using a calculator, input the base 2 and raise it to the power of 10 to find \(2^{10}\). Most scientific calculators have a function or button labeled '^' that allows you to enter this calculation directly.

Step 4: Interpret the Calculator Result

After entering the expression into the calculator, verify that the calculator displays the number 1024, which is the result of \(2^{10}\).

Key Concepts

Calculator Use

Utilizing a calculator to solve mathematical expressions, especially exponentiations, can make calculations quick and precise. When faced with an expression like \(2^{10}\), a scientific calculator will have specific buttons to facilitate exponentiation.

Here are a few steps to consider while using a calculator for exponentiation:

• Enter the base number: Input the number you want to raise to a power, which in this scenario is "2".
• Use the exponentiation function: Look for a button like "^" or a key labeled as "y^x". This button tells the calculator to treat the next number as an exponent.
• Enter the exponent: After pressing the power button, input "10", which is the exponent in our exercise.
• Calculate the result: Press the "=" or "Enter" key to compute the result, which should be 1024 in this case.

For best accuracy, ensure the numbers are input correctly and double-check your result to prevent any small errors.

Exponent Rules

Understanding exponent rules simplifies the process of working with exponential expressions. The rules provide a framework for simplifying and calculating expressions efficiently.

Key exponent rules to remember include:

• Product of Powers: When multiplying two expressions with the same base, add their exponents, e.g., \(a^m \times a^n = a^{m+n}\).
• Power of a Power: When raising an expression to another exponent, multiply the exponents, e.g., \((a^m)^n = a^{m \cdot n}\).
• Zero Exponent: Any number raised to the power of zero equals one, except when the base is zero itself, e.g., \(a^0 = 1\) for any \(a eq 0\).
• Negative Exponent: A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent, e.g., \(a^{-m} = \frac{1}{a^m}\).

In the exercise of \(2^{10}\), the use of exponent rules helps us comprehend that this involves multiplying the base, 2, by itself a total of 10 times. Hence, it's a straightforward application of the definition of what an exponent represents.

Numerical Expression Evaluation

Evaluating numerical expressions requires understanding the operations involved, especially as expressions grow complex.

In the case of \(2^{10}\), evaluating involves recognizing that exponentiation refers to repeated multiplication. Representing the base and exponent visually as \(2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\), we see the exponential operator's practical application.

Here’s a concise strategy for evaluating such expressions:

• Write out the expression as a multiplication statement to visualize the operation.
• If manually calculating, group numbers for easier multiplication (e.g., calculate \(2^5\) first and square it to get \(2^{10}\)).
• Verify results using a calculator for accuracy.

This method is efficient and helps verify the computation process, ensuring you gain a thorough understanding of how to work through exponential evaluations.