Understanding the Math Pattern

The problem presents a unique pattern using the '*' operator, which does not represent standard multiplication. We are given two examples:

• 7 * 24 = 25
• 12 * 16 = 20

Our goal is to identify the rule governing this operation and then apply it to find the value of 16 * 63.

Analyzing the Given Examples

Let's look closely at the numbers in the given examples. Do they suggest a familiar mathematical relationship?

For the first example, 7, 24, and 25 are the numbers involved. These numbers are famous because they form a Pythagorean triple. This means that the square of the largest number is equal to the sum of the squares of the other two numbers.

Let's check:

\(7^2 + 24^2 = 49 + 576 = 625\)

\(25^2 = 625\)

So, \(7^2 + 24^2 = 25^2\). This suggests that the operation \(a * b\) might be related to finding the hypotenuse \(c\) of a right-angled triangle where the other two sides are \(a\) and \(b\). The Pythagorean theorem states that \(a^2 + b^2 = c^2\), which means \(c = \sqrt{a^2 + b^2}\).

Let's test this hypothesis with the second example: 12 * 16 = 20. If the rule is \(a * b = \sqrt{a^2 + b^2}\), then:

\(12 * 16 = \sqrt{12^2 + 16^2}\)

\(12^2 = 144\)

\(16^2 = 256\)

\(12^2 + 16^2 = 144 + 256 = 400\)

\(\sqrt{400} = 20\)

This matches the given result, 20. The pattern holds true for both examples.

Applying the Pattern to 16 * 63

Now that we've identified the pattern as \(a * b = \sqrt{a^2 + b^2}\), we can apply it to the expression 16 * 63.

We need to calculate:

\(16 * 63 = \sqrt{16^2 + 63^2}\)

First, let's calculate the squares of 16 and 63:

• \(16^2 = 16 \times 16 = 256\)
• \(63^2 = 63 \times 63\)

Let's calculate \(63 \times 63\):

Or using multiplication:

  63
x 63
----
  189 (63 * 3)
3780 (63 * 60)
----
3969

So, \(63^2 = 3969\).

Now, add the squares:

\(16^2 + 63^2 = 256 + 3969\)

\(256 + 3969 = 4225\)

Finally, find the square root of the sum:

\(16 * 63 = \sqrt{4225}\)

To find \(\sqrt{4225}\), we can look for a number that, when multiplied by itself, equals 4225. Since the number ends in 5, its square root must end in 5. We can estimate:

• \(60^2 = 3600\)
• \(70^2 = 4900\)

The number 4225 is between 3600 and 4900, so its square root is between 60 and 70. The only number ending in 5 in this range is 65. Let's check 65 squared:

\(65^2 = 65 \times 65\)

  65
x 65
----
  325 (65 * 5)
3900 (65 * 60)
----
4225

So, \(\sqrt{4225} = 65\).

Conclusion

Following the identified pattern, 16 * 63 is equal to 65.

The pattern uses the concept of the Pythagorean theorem, where the operation results in the hypotenuse of a right triangle formed by the two given numbers as sides.

Revision Table: Math Pattern Analysis

Additional Information: Pythagorean Triples and Math Patterns

A Pythagorean triple is a set of three positive integers a, b, and c, such that \(a^2 + b^2 = c^2\). The most famous triple is (3, 4, 5), because \(3^2 + 4^2 = 9 + 16 = 25 = 5^2\).

The examples in this problem (7, 24, 25) and (12, 16, 20) are also Pythagorean triples. (12, 16, 20) is actually a multiple of (3, 4, 5) since \(12 = 4 \times 3\), \(16 = 4 \times 4\), and \(20 = 4 \times 5\). The triple (16, 63, 65) is another example of a Pythagorean triple.

Problems involving math patterns often require identifying the underlying mathematical relationship or rule that connects the given numbers. This could involve basic arithmetic operations, powers, roots, sequences, or geometric concepts like the Pythagorean theorem as seen in this question.

Recognizing common mathematical sets like Pythagorean triples can be helpful in solving such pattern-based problems quickly.