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.
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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 \( 9 \).
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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: \[ 63 + 81 = 9(7 + 9) \]
- We can express \( 63 \) and \( 81 \) in terms of their GCF:
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Simplifying the Expression:
- Calculate \( 7 + 9 \): \[ 7 + 9 = 16 \]
- Therefore, we have: \[ 63 + 81 = 9 \times 16 \]
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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 \( 9 \times 16 \).
- Now we can check our answer by evaluating both forms of our expression:
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, \( 144 \).