Question

Write each number in scientific notation. 0.013

Short answer

1.3 \times 10^{-2}

Step-by-step solution

Understand Scientific Notation

Scientific notation expresses numbers as a product of two factors: a coefficient (typically between 1 and 10) and a power of 10. Formally, a number is written as: \[ a \times 10^n \] where \( 1 \leq |a| < 10 \) and \( n \) is an integer.

Identify the Coefficient

Identify the coefficient by moving the decimal point in 0.013 so that it is a number between 1 and 10. Moving the decimal point two places to the right, 0.013 becomes 1.3.

Determine the Exponent

Since the decimal point was moved two places to the right, the exponent for the power of 10 will be -2 (because we moved it to the right).

Write the Number in Scientific Notation

Combine the coefficient and the power of 10. Therefore, 0.013 in scientific notation is: \[ 1.3 \times 10^{-2} \]

Key concepts

Coefficient

The coefficient is a key component in scientific notation. It's the number that you'll multiply by the power of 10.
In scientific notation, the coefficient must be a number between 1 and 10.
Let's break it down with an example: if you have the number 0.013 and need to make it manageable, you identify the coefficient by moving the decimal point.
For 0.013, moving the decimal two places to the right gives us 1.3, which falls between 1 and 10. Therefore, 1.3 is our coefficient.
Understanding this step is essential because it ensures the scientific notation forms are standardized. It's the starting point to simplify large or small numbers easily.

Power of 10

The power of 10 in scientific notation expresses how many places the decimal point has moved.
It indicates the scale of the number, whether it's large or small.
If we move the decimal point to the right, the power is negative. If it moves to the left, the power is positive.
In our example with 0.013, the decimal point moved two places to the right to turn it into 1.3. Therefore, the power of 10 becomes \(-2 \).
This tells us the original number is small, and helps in understanding its magnitude without dealing with cumbersome decimals.

Exponent

The exponent in scientific notation is the number that is placed above the 10.
It represents how many times the number 10 is multiplied by itself.
For example, if the exponent is 3, we have \[10^3 = 10 \times 10 \times 10 \].
In the given problem, 0.013 changes to 1.3 when the decimal point moves two places to the right. Thus, the exponent becomes \(-2 \) because the decimal point moved right.
This exponent helps simplify how we express and work with very large or small numbers without the need to count zeroes.