Math Pattern: 2^6 x 2^6 = 2^11 + 2^11 and 3^8 x 3^8 = 3^15 + 3^15 + 3^15

Published on August 31, 2024

Question

Okay, let’s analyze the given examples and try to produce more such examples and generalize the pattern.

Given Examples:

Understanding the Pattern:

Notice that in both examples:

We are multiplying a number raised to a power by itself (squaring it).
The result is the sum of the same number raised to a different power, repeated a certain number of times.

Let’s break down the first example:

So,

. The pattern holds.

Let’s analyze the second example:

Again,

. The pattern holds.

Producing More Examples:

Let’s try with base 5 and exponent 4:

We need to find a power of 5 and a number of repetitions such that their sum equals . Let’s try repeated 5 times:

So, we have another example:

Generalization:

Let’s generalize the pattern using variables:

Let the base be ‘a’ and the exponent be ‘n’.

Then,

We need to find a power ‘m’ and a number of repetitions ‘a’ such that:

This implies that

, so

Generalized Form:

In words:

The square of a number raised to a power ‘n’ is equal to the sum of that number raised to the power ‘2n-1’, repeated ‘a’ (the base) times.