Problem 64 - Algebra 1

Question

MULTIPLE CHOICE Which number is not in scientific notation? $$ \begin{array}{lllll} \text { (A) } 1 \times 10^{4} & \text { (B) } 3.4 \times 10^{-3} & \text { (C) } 9.02 \times 10^{2} & \text { (D) } 12.25 \times 10^{-5} \end{array} $$

Step-by-Step Solution

Answer

The number that is not in scientific notation is \(12.25 \times 10^{-5}\). Therefore, the answer is D.

Step 1: Identify the form of scientific notation

Scientific notation implies a number has one digit before the decimal point which is not 0. This number is multiplied by 10 raised to an exponent which can be any integer.

Step 2: Evaluate option A

In choice A, \(1 \times 10^{4}\) is in scientific notation as it meets the above criteria.

Step 3: Evaluate option B

In choice B, \(3.4 \times 10^{-3}\) also adheres to scientific notation as it meets the criteria mentioned.

Step 4: Evaluate option C

In choice C, \(9.02 \times 10^{2}\) is in scientific notation. It meets the criteria of scientific notation.

Step 5: Evaluate option D

In choice D, \(12.25 \times 10^{-5}\) has two numbers before the decimal point. This does not meet the criteria for scientific notation as there should only be one non-zero digit before the decimal point.

Key Concepts

Multiplying Exponents

When working with scientific notation, you often encounter expressions that require multiplying exponents. Here's a refresher on how multiplying exponents works. Suppose you have two terms in the form of scientific notation, like \( a \times 10^m \) and \( b \times 10^n \). To multiply them, follow these steps:

• Multiply the coefficients \(a\) and \(b\) normally, without considering the exponents of 10. This means just multiply the numbers themselves.
• For the 10 parts, use the property of exponents which states that when you multiply two powers with the same base, you add the exponents. So the term from multiplying \(10^m\) and \(10^n\) becomes \(10^{m+n}\).
• Combine both results to get the expression: \( (a\times b) \times 10^{m+n} \).

Let’s see an example. Multiply \( (2.5 \times 10^3) \times (4 \times 10^2) \):
- Multiply the coefficients: \(2.5 \times 4 = 10\).
- For the base 10, add exponents: \(10^3 \times 10^2 = 10^{3+2} = 10^5\).
- The product is \(10 \times 10^5\).
Always remember to check if your final answer is in the correct scientific notation format, which means making sure the coefficient is between 1 and 10.

Decimal Numbers

Decimal numbers are a core component when dealing with scientific notation. Understanding decimals will help in identifying whether a number is in the correct scientific notation format. A decimal number is a way of representing a non-whole number using a period, known as a decimal point. For example, 3.4 or 9.02.
To ensure a number is in scientific notation, it should follow certain rules:

• It must have only one non-zero digit before the decimal point, making it something between 1 and 9, like 1, 3.4, or 9.02 in scientific notation.
• The number following the decimal point can have any amount of digits, but if the number before the decimal point is not between 1 and 9, it's not in correct scientific notation.

In the original exercise, one option was incorrect because it had more than one digit before the decimal. The expression \(12.25\) in \(12.25 \times 10^{-5}\) breaks the rule because it starts with 12 instead of a number between 1 and 9. Remember, to correct such a number, you would convert it by adjusting the coefficient to a number between 1 and 9 and adjusting the exponent accordingly.

Integer Exponents

Integer exponents are numbers that tell us how many times to multiply a base number by itself. In scientific notation, the base is typically 10, and the exponent is an integer.
Here’s how integer exponents work:

• If the exponent is positive, like in \(10^3\), it means you multiply 10 by itself 3 times: \(10 \times 10 \times 10 = 1000\).
• If the exponent is negative, like in \(10^{-3}\), it represents the reciprocal of 10 raised to the absolute value of the exponent: \(1/(10^3) = 0.001\).
• This ability to represent very large or very small numbers compactly makes integer exponents in scientific notation incredibly useful in mathematics and science.

This helps when converting numbers into scientific notation. For example, converting 5000 into \(5 \times 10^3\) uses the principle of positive integer exponents, while converting 0.004 into \(4 \times 10^{-3}\) uses negative integer exponents, illustrating how 10 is scaled down.