Problem Analysis:
The task involves simplifying expressions using the rules of exponents. Specifically, we need to apply the rule for multiplying powers with the same base:
$$
a^m \cdot a^n = a^{m+n}
$$
The problem provides two expressions to simplify:
1. $ x^4 \cdot x^6 $
2. $ x^3 \cdot x^8 $
Step-by-Step Solution:
#### Expression 1: $ x^4 \cdot x^6 $
1.
Identify the bases and exponents:
- The base is $ x $.
- The exponents are $ 4 $ and $ 6 $.
2.
Apply the exponent rule:
According to the rule $ a^m \cdot a^n = a^{m+n} $, we add the exponents:
$$
x^4 \cdot x^6 = x^{4+6}
$$
3.
Simplify the sum of the exponents:
$$
4 + 6 = 10
$$
4.
Write the final answer:
$$
x^4 \cdot x^6 = x^{10}
$$
#### Expression 2: $ x^3 \cdot x^8 $
1.
Identify the bases and exponents:
- The base is $ x $.
- The exponents are $ 3 $ and $ 8 $.
2.
Apply the exponent rule:
According to the rule $ a^m \cdot a^n = a^{m+n} $, we add the exponents:
$$
x^3 \cdot x^8 = x^{3+8}
$$
3.
Simplify the sum of the exponents:
$$
3 + 8 = 11
$$
4.
Write the final answer:
$$
x^3 \cdot x^8 = x^{11}
$$
Final Answers:
1. $ x^4 \cdot x^6 = \boxed{x^{10}} $
2. $ x^3 \cdot x^8 = \boxed{x^{11}} $
Explanation:
The key concept here is the rule for multiplying powers with the same base. When multiplying such powers, you simply add the exponents while keeping the base unchanged. This rule is derived from the fundamental definition of exponents, which represents repeated multiplication. For example:
- $ x^4 $ means $ x \cdot x \cdot x \cdot x $,
- $ x^6 $ means $ x \cdot x \cdot x \cdot x \cdot x \cdot x $,
- Multiplying them together gives $ x^{4+6} = x^{10} $.
This method ensures consistency and simplifies complex expressions efficiently.
Thus, the final boxed answers are:
$$
\boxed{x^{10}, x^{11}}
$$
Parent Tip: Review the logic above to help your child master the concept of adding and subtracting exponents worksheet.
