Problem 85 - Algebra A Combined Function

Question

Write each number in standard form. $$ 3.3 \times 10^{-2} $$

Step-by-Step Solution

Answer

0.033

Step 1: Understanding Scientific Notation

Scientific notation expresses numbers as a product of a number (between 1 and 10) and a power of 10. In this exercise, the number is given as \(3.3 \times 10^{-2}\).

Step 2: Converting Power of 10

The exponent \(-2\) indicates that the decimal point of the number 3.3 should be moved two places to the left. When moving it to the left, you're essentially dividing by 100.

Step 3: Moving the Decimal Point

Start with the number 3.3 and move the decimal point two places to the left. Add zeros as placeholders if necessary. So, 3.3 becomes 0.033.

Step 4: Writing the Standard Form

The answer in standard decimal form after moving the decimal is 0.033.

Key Concepts

Understanding Standard Form

Standard form is a way of writing numbers to make them more manageable, especially when dealing with very large or small values. This method uses powers of ten to simplify numeral representation. For example, instead of writing small decimals explicitly, we use a standard form to show them compactly and insightfully.

• The standard form is particularly useful in scientific fields where data involves extremely large or minute numbers.
• It relies on the format where a number is expressed as a product: a coefficient and a power of ten, such as \( 3.3 \times 10^{-2} \).
• It must have a coefficient in the range of 1 to 10, together with an appropriate power of ten.

Understanding standard form allows you to interpret and convert these scientific notations into regular numbers efficiently, like converting \(3.3 \times 10^{-2}\) into 0.033.

The Significance of Powers of Ten

Powers of ten form the backbone of the scientific notation and standard form. By using powers of ten, we simplify the representation of numbers dramatically. Each power of ten fundamentally relies on the exponential notation.

• Positive powers (e.g., \(10^2\)) mean multiplication by ten repeatedly, resulting in larger numbers.
• Negative powers (e.g., \(10^{-2}\)) imply division by ten, which moves the decimal places to the left, producing smaller numbers.

This use of powers allows us to avoid having lengthy series of zeros in decimal numbers. When you see \(10^{-2}\), it directs you to move the decimal point two places left, converting \(3.3\) into 0.033 in standard form. It simplifies calculations and enhances readability in scientific work.

Decoding Exponents

Exponents make the handling of large and tiny numbers more systematic and less error-prone. They reflect how many times a number is multiplied by itself. Understanding and working with exponents are essential for interpreting scientific notation.

• An exponent indicates the number of times to use the base in a multiplication. For example, \(10^3\) means using ten three times: \(10 \times 10 \times 10\).
• In the case of \(10^{-2}\), you're not multiplying but instead dividing, effectively moving your decimal place left by two positions.

In scientific notation like \(3.3 \times 10^{-2}\), the exponent shows the alteration required for the decimal point, converting the expression into standard form. Once you grasp what exponents represent, it's easier to transition between exponential and decimal notations.