Problem 26 - Algebra 1: Concepts and Skills

Question

Write the number in decimal form. $$ 3 \times 10^{-4} $$

Step-by-Step Solution

Answer

The number \(3 \times 10^{-4}\) in decimal form is 0.0003.

Step 1: Understand the Exponential Value

The key thing here is to understand what the 'power of ten' or exponent (-4) means. When the exponent of 10 is negative, it means we move the decimal place to the left for that many places. In this case, we move the decimal point 4 places to the left.

Step 2: Apply the Exponential Value to the Base Number

Applying the exponent -4 to the base number 3, we put 3 followed by a decimal point and then three zeros before it. This is because we are moving the decimal place four places to the left. So the decimal point ends up before the 3. We have three zeros which is one less than the value of 4 from the exponent. This gives 0.0003.

Key Concepts

Decimal Form

Decimal form is a way of writing numbers that utilizes a decimal point to separate the whole part of a number from its fractional part. This form is used to represent numbers that are not whole, which can be particularly helpful when dealing with fractions or large numbers.

When converting a number from scientific notation like \(3 \times 10^{-4}\) to decimal form, the goal is to determine how many places to move the decimal point. In this instance, we are moving the decimal point four places to the left, based on the exponent of -4. This results in a much smaller number: 0.0003.

• The number 3 represents the base or the significant figure.
• The decimal point shift determines how the base number turns into a smaller value.
• Adding zeros is necessary if moving the decimal exceeds the number’s digits.

This process helps in visualizing very small values efficiently using decimals.

Exponents

Exponents are a mathematical notation indicating the number of times a number, known as the base, is multiplied by itself. In scientific notation, exponents are especially useful for dealing with extremely large or small numbers.

For example, with \(3 \times 10^{-4}\), the number -4 is the exponent. It tells us that we need to move the decimal point four places. Importantly, a negative exponent means the decimal point moves to the left, indicating a smaller value. Conversely, a positive exponent would mean the decimal moves to the right, creating a larger number.

• The exponent reflects the power of 10 used in calculations.
• Negative exponents are points leftward in the decimal, while positive ones point rightward.

Exponents simplify the process of expressing and computing vast or diminutive values by using powers of ten.

Powers of Ten

Powers of ten are utilized to express numbers in a compressed form, making them easier to read and work with. Understanding powers of ten involves grasping how many times 10 is multiplied or divided. This concept is foundational in scientific notation, used by scientists and mathematicians alike to manage numbers more effectively.

In the expression \(3 \times 10^{-4}\), 10 is raised to the power of -4. This indicates that we are considering one tenth, four times over, thus explaining why the decimal shifts four places left resulting in 0.0003.

• Powers of ten enable transformation of numbers to a readable form.
• This approach is particularly advantageous for very large or small numbers.
• They simplify calculations by representing numbers in a cleaner format.

This method is incredibly useful, facilitating easier computations and clarity in numerical representation.