When doing exponents, the work observed in a learner's notobook was as follows:
`4^3xx4^2=4^5`
`6^4xx6^4=6^8`
`7^3xx3^7=21^(10)`
The learner has not understood how to

Options

A. multiply numbers with same base
B. multiply, numbers with different bases
C. add exponents
D. add exponents and multiply

Answer

To solve the problem, we need to analyze the learner's notebook and identify the mistakes made in the exponent calculations. Let's break down the observations step by step.

### Step 1: Understanding the Exponent Rules
When multiplying numbers with the same base, the rule states that you should add the exponents. For example, \( a^m \times a^n = a^{m+n} \).

### Step 2: Analyzing the First Example
The learner wrote:
\[ 4^3 \times 4^2 = 4^5 \]
This is correct because:
\[ 4^3 \times 4^2 = 4^{3+2} = 4^5 \]

### Step 3: Analyzing the Second Example
The learner wrote:
\[ 6^4 \times 6^4 = 6^8 \]
This is also correct because:
\[ 6^4 \times 6^4 = 6^{4+4} = 6^8 \]

### Step 4: Analyzing the Third Example
The learner wrote:
\[ 7^3 \times 3^7 = 21^{10} \]
This is incorrect. The correct approach should involve recognizing that \( 7^3 \) and \( 3^7 \) have different bases, and thus cannot be combined using exponent rules for the same base.

### Step 5: Identifying the Mistake
The learner mistakenly added the bases instead of recognizing that they cannot be combined:
- The correct interpretation should be that \( 7^3 \) and \( 3^7 \) cannot be multiplied directly to form \( 21^{10} \).

### Conclusion
The learner has not understood how to multiply numbers with different bases. The correct answer is option A: multiply numbers with the same base.