Problem 45 - Fundamentals of Mathematics

Question

Determine the value of each of the powers. Use a calculator to check each result. \(9^{3}\)

Step-by-Step Solution

Answer

The value of \(9^3\) is 729.

Step 1: Understand the Problem

We need to calculate the value of the power expression \(9^3\). This expression means that we are multiplying the base number 9 by itself three times.

Step 2: Set Up the Power Calculation

Write the expression \(9^3\) as a multiplication: \(9 \times 9 \times 9\). We will first multiply to find \(9 \times 9\), and then multiply the result by 9 once more.

Step 3: Calculate the First Multiplication

Multiply the first two numbers: \(9 \times 9 = 81\).

Step 4: Calculate the Final Result

Now multiply the result from Step 3 by 9: \(81 \times 9\).

Step 5: Perform the Second Multiplication

Performing the multiplication, we find \(81 \times 9 = 729\).

Step 6: Verify with a Calculator

As a confirmation, use a calculator to compute \(9^3\). The result should also be 729.

Key Concepts

powers

Powers are a way of expressing numbers that are multiplied by themselves multiple times. In an expression like \(9^3\), the number 9 is called the base, and 3 is the exponent. The exponent indicates how many times the base is used in a multiplication. Therefore, \(9^3\) means you multiply 9 by itself three times: \(9 \times 9 \times 9\). This process is also known as exponentiation, a fundamental concept in mathematics that helps simplify repeated multiplication.

• Base: The number being multiplied.
• Exponent: Indicates how many times to multiply the base by itself.
• Exponentiation: The operation of raising one number to the power of another.

This concept is very useful in various fields such as physics, engineering, and computer science, where powers can help solve complex calculations quickly by reducing repetitive work.

multiplication

Understanding multiplication is key when dealing with powers. When you are given a power expression like \(9^3\), you can break it down into simpler multiplication steps. Let's explore the example from our exercise.

First, recognize that \(9^3\) means multiplying 9 by itself a total of three times: \(9 \times 9 \times 9\). This approach allows you to carry out the multiplication in parts:

• First, calculate \(9 \times 9 = 81\).
• Then, use the result 81 and multiply it by 9: \(81 \times 9 = 729\).

By breaking it into smaller steps, the complexity is reduced, and it becomes easier to ensure that each multiplication is performed accurately. Multiplication, especially by hand, can be prone to small errors, so careful, step-by-step calculation is important for accuracy.

calculation verification

Verification is an important aspect of solving mathematical problems. Once you've calculated \(9^3 = 729\) through manual multiplication, it's essential to check your work. Calculation verification ensures accuracy and helps to avoid errors in any mathematical task.

Using a calculator is a quick and effective method to verify your result.

• Re-enter your original power value \(9^3\) into a calculator.
• Compare the calculator’s output with your manual calculation; it should match the number 729.
• If the numbers don’t match, double-check each step of your manual calculation to find where you may have gone wrong.

Verification not only confirms correctness but also builds confidence in your problem-solving skills. In academic settings and real-life applications, being adept at verifying calculations is crucial.