how do you multiply two scientific numbers together?

When multiplying two numbers in **scientific notation**, follow these steps:

### **General Form of Scientific Notation**
A number in scientific notation is written as:

\[
a \times 10^b
\]

where:
- \( a \) is a number between **1 and 10** (\( 1 \leq |a| < 10 \)),
- \( 10^b \) is a power of ten.

### **Steps to Multiply Two Scientific Notation Numbers**
Suppose you have:

\[
(A \times 10^m) \times (B \times 10^n)
\]

1. **Multiply the coefficients** (\( A \) and \( B \)):  
   \[
   A \times B
   \]
2. **Add the exponents** of 10 (\( m \) and \( n \)):  
   \[
   10^{m+n}
   \]
3. **Adjust if necessary**:  
   If the new coefficient is **not between 1 and 10**, adjust it by moving the decimal and modifying the exponent.

---

### **Example 1**
Multiply \( (3.2 \times 10^4) \times (5.0 \times 10^6) \).

1. Multiply the coefficients:  
   \[
   3.2 \times 5.0 = 16.0
   \]
2. Add the exponents:  
   \[
   10^{4+6} = 10^{10}
   \]
3. Adjust the coefficient:  
   Since **16.0** is **not** between 1 and 10, rewrite it as **\( 1.6 \times 10^1 \)**.

4. Multiply by \( 10^{10} \):  
   \[
   (1.6 \times 10^1) \times 10^{10} = 1.6 \times 10^{11}
   \]

**Final Answer:**  
\[
1.6 \times 10^{11}
\]

---

### **Example 2**
Multiply \( (6.0 \times 10^{-3}) \times (2.0 \times 10^5) \).

1. Multiply the coefficients:  
   \[
   6.0 \times 2.0 = 12.0
   \]
2. Add the exponents:  
   \[
   10^{-3+5} = 10^2
   \]
3. Adjust the coefficient:  
   Since **12.0** is **not** between 1 and 10, rewrite it as **\( 1.2 \times 10^1 \)**.

4. Multiply by \( 10^2 \):  
   \[
   (1.2 \times 10^1) \times 10^2 = 1.2 \times 10^3
   \]

**Final Answer:**  
\[
1.2 \times 10^3
\]

---

Would you like more examples or practice problems? Here are some related questions:

1. How do you divide numbers in scientific notation?
2. How do you convert a number into scientific notation?
3. How do you add or subtract numbers in scientific notation?
4. Why do we use scientific notation in real-world applications?
5. How do you handle negative exponents in scientific notation?

**Tip:** Always check if your final coefficient is between 1 and 10. If not, adjust it and modify the exponent accordingly.

Learn the process of multiplying two numbers in scientific notation with detailed examples. This guide covers how to multiply coefficients, add exponents, and adjust the final result to maintain proper scientific notation. Ideal for students in Grades 8-10.

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