Problem 7 - Algebra 2 and Trigonometry

Question

Write each number as a power. \(1,000\)

Step-by-Step Solution

Answer

1,000 can be written as \(10^3\).

Step 1: Recognize base and exponent

Notice that the number 1,000 can be expressed as a power of 10. We need to find a base number that, when raised to an exponent, gives us 1,000.

Step 2: Express 1,000 as a product of tens

Break down the number 1,000 into a repeated multiplication involving the number 10. That would look like this: \[ 1,000 = 10 \times 10 \times 10 \]This shows that 1,000 is composed of three 10s multiplied together.

Step 3: Determine the power representation

Since multiplying three 10s together equates to 1,000, we can express it as a power: \[ 10^3 = 1,000 \]Here, 10 is the base and 3 is the exponent.

Key Concepts

Power of a number

A power of a number is a way to express repeated multiplication of the same number by itself. It's a convenient way to write large numbers or perform calculations more easily. For example, the power symbol, an exponent, lets us represent a large multiplication in a compact form.
When you see a number like \(10^3\), it reads as "10 raised to the power of 3," meaning 10 is multiplied by itself three times. Rather than writing 10 multiplied three times like \(10 \times 10 \times 10\), we simplify it to a power. This makes handling large numbers and equations much more manageable.
Benefits of using powers include:

• Providing a simplified notation for writing big products.
• Making it easier to calculate with big numbers.
• Helping us to understand the scale and size of numbers very quickly.

Recognizing and using the powers of numbers are fundamental skills, particularly in scientific disciplines, engineering, and in dealing with exponential growth.

Base and exponent

The terms "base" and "exponent" are crucial when working with powers. The base is the number that is being multiplied, while the exponent indicates how many times the base is used in the multiplication. Let's take an in-depth look:

• Base: In the expression \(10^3\), the number 10 is the base. It's the main number that gets multiplied.
• Exponent: Here, the number 3 is the exponent. It tells us that the base (10) is used three times in multiplication.

The interplay between the base and exponent can change the outcome significantly. For instance, changing the base affects the size of the resulting power, whereas increasing the exponent increases the multiplication effect, magnifying the final value.
Understanding base and exponent clears up confusion when interpreting numbers written in power form, allowing for a better grasp of more advanced mathematical concepts.

Multiplication

Multiplication is one of the four basic arithmetic operations, and it plays a key role in understanding powers of numbers. When you multiply a number by itself repeatedly, you're seeing multiplication in action driving the formation of powers.
For example, to find \(10^3\), you multiply 10 by itself twice more:

• First: \(10 \times 10 = 100\)
• Second: \(100 \times 10 = 1,000\)

Multiplication here isn't just a calculation; it's the basis for understanding how an exponent affects a base.
Additionally, the multiplication of the numbers combined with the concept of powers helps in simplifying complex expressions and calculations. This foundational arithmetic operation is essential for everyday calculations and is key for tackling advanced math problems.
By being comfortable with multiplication, you can more easily transition into working effectively with powers and exponential expressions.