To use the distributive property and the greatest common factor (GCF) to find an equivalent expression for \(63 + 81\), we first need to identify the GCF of the two numbers.
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Find the GCF of 63 and 81:
- The prime factorization of \(63\) is \(3^2 \times 7\).
- The prime factorization of \(81\) is \(3^4\).
- The common prime factor is \(3\), and the minimum exponent is \(2\). Hence, the GCF is \(3^2 = 9\).
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Rewrite each number in terms of the GCF:
- Now, we can express \(63\) and \(81\) in terms of the GCF: \[ 63 = 9 \times 7 \] \[ 81 = 9 \times 9 \]
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Use the distributive property:
- We can now rewrite the original expression as follows: \[ 63 + 81 = 9 \times 7 + 9 \times 9 \]
- Using the distributive property, we can factor out the \(9\): \[ 63 + 81 = 9 \times (7 + 9) \]
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Simplify the expression:
- Now simplify inside the parentheses: \[ 7 + 9 = 16 \]
- Therefore, we have: \[ 63 + 81 = 9 \times 16 \]
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Equivalent expression:
- The equivalent expression for \(63 + 81\) using the GCF and the distributive property is: \[ 9 \times 16 \]
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Check the answer:
- To check our answer, we will compute both the original expression and the equivalent one: \[ 63 + 81 = 144 \] \[ 9 \times 16 = 144 \]
- Since both calculations yield the same result, \(144\), we have confirmed that our equivalent expression is correct.
In summary, by using the GCF of \(63\) and \(81\), we factored the expression to \(9 \times (7 + 9)\) and found that \(63 + 81\) is equivalent to \(9 \times 16\). We verified our work by calculating both sides, confirming they yield the same result.