Problem 5 - Algebra 2 and Trigonometry

Question

Write each number as a power. 25

Step-by-Step Solution

Answer

25 can be written as \( 5^2 \).

Step 1: Identify the Base and Exponent

To express 25 as a power, identify a base number that when raised to an exponent gives 25. Since 25 is a perfect square, think about squares of smaller whole numbers.

Step 2: Calculate the Square

Consider smaller whole numbers as bases. For base 5, calculate: \[ 5^2 = 5 \times 5 = 25 \]

Step 3: Express 25 as a Power

Since squaring 5 results in 25, we can express 25 as a power of 5: \[ 25 = 5^2 \]

Key Concepts

Perfect Squares

A perfect square is a number that can be expressed as the square of an integer. In simple terms, it's what you get when you multiply a number by itself. For example, 1, 4, 9, 16, and 25 are all perfect squares. Understanding perfect squares can help simplify calculations and recognize patterns in numbers.
When you see a number, thinking about whether it's a perfect square can be helpful. It's like knowing shortcuts in math that make problem-solving easier. In the case of 25, recognizing it as a perfect square can lead you directly to the fact that it is the result of squaring 5.

Base and Exponent

The terms 'base' and 'exponent' are crucial when working with powers. A power is expressed in the form of a base raised to an exponent. It looks like this:

• Base: The number that is being multiplied by itself.
• Exponent: Tells how many times the base is used in the multiplication.

For example, in \( 5^2 \), 5 is the base, and 2 is the exponent. This tells you to multiply 5 by itself once (since 5 appears twice in the multiplication 5 x 5). This notation is useful because it simplifies the way you write repeated multiplication. Instead of writing 5 x 5, you can simply write \( 5^2 \).This concept can help you deal with larger numbers, too. Knowing the base and the exponent allows you to quickly understand the size and nature of a number.

Powers of Numbers

Understanding powers involves recognizing the relationship between numbers through multiplication. When you use powers, you're essentially multiplying a number by itself a specific number of times. This can greatly simplify calculations and help you grasp complex math concepts with ease.Let's use the example of 25 again to see powers in action. When expressed as \( 5^2 \), it shows that you multiply 5 by itself to get 25. Each power has its own unique base and exponent, and it’s these two elements that define the resultant value of the power.Using powers can make math more understandable and less intimidating. Whether dealing with scientific notation, calculating areas, or exploring algebraic expressions, mastering the concept of powers gives you a powerful tool for handling numbers efficiently. Recognizing powers also allows for quick mental arithmetic and helps simplify calculations that would otherwise require more complex steps.