Standard form is a method of writing very large or very small numbers succinctly, commonly used in scientific, mathematical, and engineering contexts. It simplifies numbers to a format of $A \times 10^n$, where $A$ is a coefficient between 1 and 10 (including 1 but not 10), and $n$ is an integer indicating the power of 10.

**What is Standard Form?**

The standard form notation $A \times 10^n$ allows us to express numbers in a more manageable way, especially when dealing with extremely large or small values.

**Key Concepts**

**Coefficient**$(A)$: A number between 1 and 10.**Power of 10**$(10^n)$: Indicates how many places the decimal point has been moved. $n$ can be positive, negative, or zero.

**Converting Numbers into Standard Form**

Converting a number into standard form involves two steps:

**1. Find the Coefficient**: Adjust the decimal point so there is only one non-zero digit to the left.

**2. Determine the Power of 10**: Count the decimal moves. This count, $n$, is positive if moved left, negative if right.

**Example 1: Large Number**

**Original**: 123,456**Standard Form**: $1.23456 \times 10^5$

**Example 2: Small Number**

**Original**: 0.00789**Standard Form**: $7.89 \times 10^{-3}$

**Converting Numbers Out of Standard Form**

To convert from standard form to a normal number:

**1. Move the Decimal Point**: Based on $n$, move the decimal point right (if $n$ is positive) or left (if $n$ is negative).

**2. Adjust with Zeros**: Add zeros as needed to fill gaps.

**Example 1: From Standard Form**

**Standard Form**: $3.65 \times 10^4$**Normal Number**: 36,500

**Example 2: From Standard Form**

**Standard Form**: $4.2 \times 10^{-3}$**Normal Number**: 0.0042

**Practice Questions**

1. Convert 5,000,000 into standard form.

**Solution:**

2. Convert $8.2 \times 10^6$ into a normal number.

**Solution:**

3. Calculate $2.5 \times 10^3 \times 4 \times 10^2$ in standard form.

**Solution:**