Question

Consider the expression 63 + 81.
How can you use the distributive property and the GCF to find an equivalent expression? Explain how you can check your answer.

Answer 1

To use the distributive property and the greatest common factor (GCF) to find an equivalent expression for \( 63 + 81 \), we start by identifying the GCF of the two numbers.

1. **Finding the GCF**:
- The prime factorization of \( 63 \) is \( 3^2 \times 7 \) (since \( 63 = 3 \times 21 = 3 \times 3 \times 7 \)).
- The prime factorization of \( 81 \) is \( 3^4 \) (since \( 81 = 3 \times 27 = 3 \times 3 \times 9 = 3 \times 3 \times 3 \times 3 \)).
- The common prime factor is \( 3 \). The lowest power of \( 3 \) that appears in both factorizations is \( 3^2 \).
- Thus, the GCF of \( 63 \) and \( 81 \) is \( 9 \).

2. **Applying the Distributive Property**:
- We can express \( 63 \) and \( 81 \) in terms of their GCF:
- \( 63 = 9 \times 7 \)
- \( 81 = 9 \times 9 \)
- Using these forms, we can rewrite the original expression:
\[
63 + 81 = 9 \times 7 + 9 \times 9
\]
- Now, we apply the distributive property:
\[
63 + 81 = 9(7 + 9)
\]

3. **Simplifying the Expression**:
- Calculate \( 7 + 9 \):
\[
7 + 9 = 16
\]
- Therefore, we have:
\[
63 + 81 = 9 \times 16
\]

4. **Checking Your Answer**:
- Now we can check our answer by evaluating both forms of our expression:
- First, directly calculate \( 63 + 81 \):
\[
63 + 81 = 144
\]
- Next, calculate \( 9 \times 16 \):
\[
9 \times 16 = 144
\]
- Since both calculations yield the same result, \( 144 \), we have confirmed that \( 63 + 81 \) is indeed equivalent to \( 9(7 + 9) \) or \( 9 \times 16 \).

In conclusion, using the distributive property and the GCF, we found that:
\[
63 + 81 = 9(7 + 9) \quad \text{or} \quad 63 + 81 = 9 \times 16
\]
And we checked our answer by confirming both calculations produce the same result, \( 144 \).