Problem 62 - Beginning and Intermediate Algebra

Question

Convert to scientific notation. $$3.4 \times 10^{-6}$$

Step-by-Step Solution

Answer

The given number is already in correct scientific notation: \(3.4 \times 10^{-6}\).

Step 1: Verify the coefficient

Check if the coefficient \(a\) is between 1 and 10: \[1 \leq 3.4 < 10\] It is, so the coefficient is in the correct form.

Step 2: Verify the exponent

Check if the exponent is an integer: The given exponent is \(-6\) which is indeed an integer. Since both the coefficient and exponent are in their correct form, we can conclude that the given number is already in scientific notation: \[3.4 \times 10^{-6}\]

Key Concepts

Coefficients

In scientific notation, the coefficient is the number in front of the multiplication sign. It should always be between 1 and 10. This is super important because it standardizes the way we represent numbers. For example, in the expression \(3.4 \times 10^{-6}\), the coefficient is 3.4. This number tells us the essential value of the number without all the zeros exaggerating its size. This standardization helps us easily compare very large or very small numbers in a consistent manner.

Exponents

Exponents in scientific notation describe how many times the number 10 is multiplied by itself. They can be positive or negative. Negative exponents denote numbers less than one, like fractions. In our example, the exponent is \(-6\). This means our coefficient, 3.4, is divided by 10 six times. This placement of the decimal gives us a tiny number. Understanding exponents is crucial for mathematics as they simplify expressions and calculations.

Integer

An integer is a whole number that can be positive, negative, or zero. In scientific notation, exponents need to be integers. This is why \(-6\) in \(3.4 \times 10^{-6}\) is an integer. This ensures the operation – multiplying by powers of ten – is simple. The use of integer exponents keeps calculations straightforward and numbers easy to understand. When you see an integer in scientific notation, you know exactly how many places the decimal will move.

Mathematics Education

Mathematics education often emphasizes concepts like scientific notation because they play a vital role in practical scenarios. For example, scientific notation is key in fields like astronomy and physics, where we deal with extremely large distances or very small particles. This notation simplifies calculations and helps students grasp complex ideas more easily. By explaining concepts like coefficients and exponents effectively, students can apply these skills in real-world problems.