Problem 4 - Algebra 2 and Trigonometry

Question

Write each number as a power. 27

Step-by-Step Solution

Answer

The number 27 can be written as \(3^3\).

Step 1: Recognize the Base

Identify the base that is raised to a power to form the number 27. The common bases to try include small integers like 2, 3, and 5.

Step 2: Experiment with Powers

Try raising the base to different powers to match the given number, 27. Check: \( 3^1 = 3 \) \( 3^2 = 9 \) \( 3^3 = 27 \)

Step 3: Find the Correct Expression

Since \(3^3 = 27\), the number 27 can be written as a power using 3 as the base and 3 as the exponent.

Key Concepts

Base and Exponent

When we talk about powers and exponents, two essential components come into play: the base and the exponent. Understanding these elements helps us express numbers more efficiently.

• Base: This is the number that gets multiplied. It is the main number in an expression, like the number 3 in the expression \(3^3\).
• Exponent: This tells us how many times to use the base in a multiplication. In \(3^3\), the exponent is 3, indicating that 3 is used three times in multiplication (\(3 \times 3 \times 3\)).

Recognizing the base and exponent is similar to recognizing familiar faces in a crowd. It simplifies the understanding of multiplication repeated multiple times. This is integral to simplifying operations and solving mathematical expressions proficiently.

Exponential Notation

Exponential notation is a shorthand way of expressing repeated multiplication of the same number. It’s like a language shortcut that turns a long multiplication sequence into a compact form.

• In exponential notation, \(b^n\), the base \(b\) is multiplied by itself \(n\) times. It's highlighted by looking at \(3^3\), which equals \(3 \times 3 \times 3\).
• This method makes it easier to perform calculations involving large numbers, and it's especially useful in fields involving mathematics, engineering, and the sciences.

Exponential notation bridges the gap between simplicity and complexity, converting lengthy multiplication into a simple form that’s easier to manipulate and understand. This notation is an essential mathematical tool.

Power of a Number

Expressing a number as a power involves writing it in the form \(b^n\). This is where you show the relationship between the base, the exponent, and the overall result. For example, the number 27 can be expressed as a power of 3: \(3^3 = 27\). This is because the base (3) is used in a multiplication three times (3 multiplied by itself twice more), resulting in the number 27.

• To understand the power of a number, break it down to see how many times the base needs to be multiplied to get the final result.
• This expression helps by simplifying complex equations and providing insight into the behavior of numbers in operations.

The concept of a power helps in recognizing patterns and simplifying calculations, forming a core part of mathematical problem-solving. It's like demonstrating the secret behind a magic trick, revealing its simplicity.