Problem 86 - Algebra A Combined Function

Question

Write each number in standard form. $$ 4.8 \times 10^{-6} $$

Step-by-Step Solution

Answer

The number in standard form is 0.0000048.

Step 1: Understand Standard Form

Standard form (or scientific notation) is a way of writing numbers that are too large or too small to be conveniently written in decimal form. It includes a coefficient that is greater than or equal to 1 and less than 10, multiplied by 10 raised to a power.

Step 2: Start with the Given Number

We are given the number in scientific notation: \(4.8 \times 10^{-6}\). This means the coefficient is 4.8 and it is multiplied by 10 to the power of -6.

Step 3: Move the Decimal Point

Move the decimal point of 4.8 six places to the left because of the negative power of ten. One move to the left changes 4.8 to 0.48, two moves to 0.048, and so on.

Step 4: Finish Conversion

After shifting the decimal point six places to the left, the number becomes 0.0000048. This is the number in standard (or decimal) form.

Key Concepts

standard form

In mathematics, the standard form is a useful way to express extremely large or small numbers. It is also known as scientific notation. The purpose is to easily handle and understand complex numbers without writing them fully out in decimal form.

• Standard form involves a number known as the coefficient.
• The coefficient must be between 1 and 10, but not equal to 10.
• It is then multiplied by 10 raised to an exponent, which tells us how many places to move the decimal to convert it back to decimal form.

For example, if we take the number 4.8 and express it as 4.8 \(\times\) 10^{-6}, the first part (4.8) is the coefficient and 10^{-6} indicates that the decimal needs to move six places to the left.
Understanding standard form helps us write very big or very small numbers in a clear and concise way that makes calculations easier.

decimal form

Decimal form is the way of writing numbers using a decimal point. It is what people typically use in day-to-day life to express figures like money, weights, and measures.

• In decimal form, each digit after the decimal represents tenths, hundredths, thousandths, and so on.
• This form makes the number intuitive to understand and operate with.

When we convert from standard form like 4.8 \(\times\) 10^{-6} to decimal form, we move the decimal to the left six times, resulting in 0.0000048. This makes the number legible, and we can easily see how "small" or "big" it actually is.
Understanding decimal form helps bridge the gap between scientific notation (standard form) and everyday arithmetic, providing clarity in interpretation.

negative exponents

Negative exponents are an important concept when working with numbers in standard form. An exponent tells you how many times to multiply a number by 10. A negative exponent means you are dividing by 10 each time instead of multiplying.

• A negative exponent moves the decimal to the left.
• The size of the number decreases as the number of decimal places increases.

For the number 4.8 \(\times\) 10^{-6}, the -6 indicates that you move the decimal six places to the left. This process transforms the number from its standard form to decimal form: 0.0000048.
Handling negative exponents can initially be challenging but remember, they help in denoting small numbers efficiently, and mastering them provides a better understanding of number scales and their impact.