Problem 5 - Astronomy: At Play in the Cosmos

Question

Consider a number \((x)\) that is 4 orders of magnitude larger than another \((y) .\) Which accurately describes the value of \(x\) compared to \(y\) ? a. It is 4 times larger. d. It is 10,000 times larger. b. It is 10 times larger. e. It is 400 times larger. c. It is 1,000 times larger.

Step-by-Step Solution

Answer

x is 10,000 times larger than y.

Step 1: Understand Orders of Magnitude

Orders of magnitude refer to the power of ten by which numbers differ. One order of magnitude difference means multiplying by 10. Hence, if something is 4 orders of magnitude larger, it means the number is multiplied by 10 four times.

Step 2: Calculate 4 Orders of Magnitude

Given that 4 orders of magnitude larger means multiplying by 10 four times, calculate this as follows: multiply 10 by itself 4 times: \(10^4\).

Step 3: Compute the Value of \(10^4\)

Calculate \(10^4\), which is \(10 \times 10 \times 10 \times 10\), resulting in 10,000.

Step 4: Select the Correct Option

The computation in the previous step shows that \(x\) is 10,000 times larger than \(y\). The option that describes this accurately is option (d), which states it is 10,000 times larger.

Key Concepts

Understanding Power of Ten

When we talk about the "power of ten," we're really discussing how many times a number is multiplied by 10. This is an essential concept in mathematics and science because it allows us to easily understand and represent large numbers.

For example, rather than writing 1,000,000, we can express it as a power of ten:

• The number 10 multiplied by itself 6 times is represented as \(10^6\).
• So, \(10^6 = 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 1,000,000\).

Every time we multiply a number by 10, we increase its power of ten by one. This means if something is described as being a higher power of ten, it has more zeros attached to it, making it exponentially larger.

The Role of Scientific Notation

Scientific notation is a method to express very large or very small numbers in a concise form without losing precision. It's especially handy when dealing with extreme values in a wide range of fields like physics, chemistry, or engineering.

This notation represents numbers as a product of two factors:

• A decimal number (usually between 1 and 10), and
• An exponential factor, which is a power of ten.

For instance, the number 500,000 can be expressed in scientific notation as \(5 \times 10^5\). This clearly indicates that the original number is 5 followed by five zeros.

Scientific notation not only simplifies the expression of numbers but also makes arithmetic operations like multiplication and division more manageable.

Exponents Explained

Exponents are a fundamental mathematical concept that describes how many times a number, known as the base, is multiplied by itself. Understanding exponents is crucial for grasping the idea of powers of ten and scientific notation.

In exponential notation:

• The base is the number being multiplied, and
• The exponent tells us how many times it is multiplied.

For example, in \(3^4\), 3 is the base, and the exponent 4 means that 3 is multiplied by itself four times:

• \(3 \times 3 \times 3 \times 3 = 81\)

When using powers of ten, the base is always 10. The exponent describes the number of times 10 is used as a factor. The larger the exponent, the more "zeros" the resulting number will have. This helps in quickly manipulating and understanding large numbers.