Problem 39 - Algebra 1: Concepts and Skills

Question

Use a calculator to evaluate the power. $$ 3^{12} $$

Step-by-Step Solution

Answer

The result of \(3^{12}\) is 531441.

Step 1: Understand exponential notation

An expression like \(3^{12}\) is in exponential notation. This expresses the operation of raising a number, in this case 3, to a power, here 12. It means to multiply 3 by itself 12 times.

Step 2: Input expression into the calculator

Input the base number, 3, then use the button that corresponds to exponential notation, which typically looks like '^', then input the exponent, 12.

Step 3: Evaluate the expression

Press the equals button (=), or whatever button your calculator uses to compute results. The calculator will perform the operation and provide the result.

Key Concepts

Exponential Notation

When we refer to exponential notation, we are talking about a mathematical way to show that a number is being multiplied by itself a certain number of times. This is commonly seen in the format of \(a^b\), where \(a\) is called the base, and \(b\) is called the exponent or power. For example, in the expression \(3^{12}\), the base is 3, and the exponent is 12. Here, the number 3 is multiplied by itself 12 times.

Exponential notation is a compact way of expressing repeated multiplication.

• It simplifies expressions, making them easier to read and manage.
• It is widely used in various areas of mathematics and science.
• It forms the basis for scientific notation, which expresses large numbers more succinctly.

By mastering exponential notation, calculations with large powers become much more manageable.

Using a Calculator

Using a calculator to handle exponential expressions can make the process quick and straightforward. Most scientific and graphing calculators have a special button for exponentiation, often labeled as '^', 'exp', or simply 'power'. Here’s a step-by-step guide on how to use the calculator for evaluating powers:

• First, input the base value. In our example of \(3^{12}\), you would first type '3'.
• Next, look for and press the button that allows you to enter the exponent or power. This might be '^' or another similar symbol.
• Finally, input the exponent value, which is '12' in this case.
• Press the '=' button or the equivalent on your calculator to calculate the power.

Calculators simplify the process, reducing the chance of error from manual calculations. If you're ever unsure, consult the user's manual of your calculator for guidance.

Exponentiation

Exponentiation is the mathematical operation that involves raising a number (the base) to a certain power (the exponent). It is a fundamental operation in mathematics, akin to addition or multiplication, but it involves repeated multiplication.

Some important aspects of exponentiation:

• Understanding powers: When you see \(a^b\), it means multiplying the base \(a\) by itself \(b\) times. For instance, \(3^{12}\) involves multiplying 3 by itself 12 times.
• Handling large numbers: Exponentiation can quickly lead to very large numbers, as the operation grows the number exponentially.
• Usage in formulas: Exponentiation is crucial in growth models, physics formulas, and more, as it helps simplify expressions involving rapid scaling.

The knowledge of exponentiation, combined with using calculators to perform these operations, can greatly aid in tackling complex mathematical problems efficiently.