Problem 56 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Expression

The expression we need to evaluate is \(2^{10}\). This indicates that the base number, 2, is raised to the 10th power.

Step 2: Calculate the Exponentiation

Using a calculator, enter the base number 2, and then use the exponentiation function (often represented as a caret symbol or a specific button on the calculator) to raise 2 to the power of 10. The calculation will be \(2^{10} = 1024\).

Step 3: Confirm Your Result

Double-check the calculation by reviewing the calculator's input and ensuring that the correct function was used. If done correctly, you should have \(1024\).

Key Concepts

Power of a Number

Exponentiation is an essential mathematical operation where a number, called the base, is raised to an exponent (or power). This operation is expressed in the form of \(a^b\), where \(a\) is the base and \(b\) is the exponent. Raising a number to a power means multiplying the base by itself as many times as the exponent indicates. For example, \(2^{10}\) means multiplying 2 by itself 10 times, which equals 1024.

• Base: The number that is multiplied.
• Exponent: How many times to use the base in multiplication.
• Example: \(2^3 = 2 \times 2 \times 2 = 8\)

When working with powers, it’s crucial to remember that an exponent of 0 results in 1 for any non-zero base, and an exponent of 1 simply returns the base.

Calculator Usage in Mathematics

Modern calculators simplify many mathematical calculations, including exponentiation. Understanding how to use a calculator effectively is vital for efficient problem-solving. To calculate powers,

• Locate the Function: Most calculators will have a specific button for raising numbers to a power, often labeled as \(x^y\) or marked by a caret symbol \(^\wedge\).
• Input the Base: For example, press '2' for the base in \(2^{10}\).
• Use the Exponent Key: Press the power function button, then input '10' for the exponent.

Finally, confirm your input to ensure correctness. For calculator-based tasks, the response time is fast, making them ideal for checking manual calculations.

Numerical Expressions

Numerical expressions involve numbers and operations combined in a meaningful way. These expressions help to simplify and solve mathematical problems. In the case of exponentiation, the numerical expression \(2^{10}\) breaks down into specific components:

• Base: The number 2.
• Exponent: Represented by 10.
• Exponentiation: The operation of raising the base to the power of the exponent.

By following specific computational steps, numerical expressions are evaluated to find their value. Practicing with numerical expressions, such as calculating \(2^{10} = 1024\), enhances problem-solving skills in mathematics, making tasks quicker and more efficient.