Problem 13 - Prealgebra

Question

SOUND Fireworks can easily reach a sound of 169 decibels, which can be dangerous if prolonged. Write this number using a power greater than 1 and a lesser base.

Step-by-Step Solution

Answer

169 can be expressed as \(13^2\).

Step 1: Understanding the Problem

We need to express 169 as a power with a base and an exponent, where the exponent is greater than 1.

Step 2: Finding the Base and Exponent

We need to find a base number that, when raised to a certain power, equals 169. Common smaller bases to try are integers like 2, 3, 4, and so on.

Step 3: Trial with Base 13

Let's check if 13 raised to the power of 2 equals 169. We calculate: \(13^2 = 13 \times 13 = 169\).

Step 4: Verification

Since 169 equals 13 squared, our representation as a power is correct. We have found that 169 can be expressed as the power \(13^2\).

Key Concepts

Understanding Base and Exponent

In mathematics, numbers can be expressed in various forms, including using the concept of base and exponent. This expression, often referred to as exponential notation, consists of two main parts:

• The base - This is the number that is being multiplied by itself.
• The exponent - This signifies how many times the base is used as a factor in the multiplication.

For example, in the expression \(5^3\), 5 is the base, and 3 is the exponent, indicating that 5 is multiplied by itself three times: \(5 \times 5 \times 5 = 125\).
In our original problem about fireworks reaching a sound of 169 decibels, we needed to express 169 as a base and exponent, where the exponent is greater than 1. We discovered that 13 is the base, and 2 is the exponent, as \(13^2 = 169\).
This shows how understanding base and exponent can help break down numbers into manageable parts.

Exploring Mathematical Powers

The term "power" is synonymous with the idea of raising a number to an exponent. When we talk about a number raised to a power, we mean we're multiplying the number, known as the base, repeatedly according to the given power or exponent.
Mathematical powers are vital for simplifying complex calculations. Instead of writing a long product of repeated multiplications, we use a concise power notation. For example:

• \(2^4\) means 2 is raised to the 4th power, which equals \(2 \times 2 \times 2 \times 2 = 16\).
• A power of 2, like in our original problem \(13^2\), simplifies the multiplication of 13 by itself, providing an efficient expression.

Mathematical powers help in fields like algebra, physics, and computer science, as they make numbers easier to manipulate and understand at scale.
This concept is essential, as seen in how we used it to express 169 decibels in terms of powers, giving us \(13^2\).

Working with Integer Powers

Integer powers refer to the process of using whole numbers as exponents. This concept is particularly practical because working with whole numbers facilitates straightforward arithmetic and verification of calculations.
Whenever the exercises involve powers where exponents are whole numbers, such as \(n^1, n^2, n^3\), and so on, it makes computation simplified and more intuitive. This applies especially for calculations involving very large or very small numbers.

• For example, in the context of the decibel problem, we employed an integer power by setting the base as 13 and the exponent as 2.
• In this case, \(13^2 = 169\), illustrates a typical instance of employing integer powers.

Working with integer powers enables students to see practical and real-world applications of exponents, like converting a number such as 169 into an understandable form derived from multiplication.
Ultimately, using integer powers makes calculations more accessible and serves as a fundamental aspect of learning in mathematics.