Find the value of x:

(3 + 4) (3² + 4²) (3⁴ + 4⁴) (3⁸ + 4⁸) (3¹⁶ + 4¹⁶) (3³² + 4³²) = (4^x - 3^x)

We know the algebraic identity:

(a - b) (a + b) = a² - b²

The given expression can also be written as,

(4 + 3) (4² + 3²) (4⁴ + 3⁴) (4⁸ + 3⁸) (4¹⁶ + 3¹⁶) (4³² + 3³²) = (4^x - 3^x)

To use our identity, we need a (4 - 3) term at the beginning. Since 4 - 3 = 1, multiplying the left side by (4 - 3) does not change its value:

∴ Given expression can be re-written as:

(4 - 3)(4 + 3) (4² + 3²) (4⁴ + 3⁴) (4⁸ + 3⁸) (4¹⁶ + 3¹⁶) (4³² + 3³²) = (4^x - 3^x)

From the algebraic identity,

(4 - 3) (4 + 3) = (4² - 3²)

(4² - 3²) (4² + 3²) = (4⁴ - 3⁴)

(4⁴ - 3⁴) (4⁴ + 3⁴) = (4⁸ - 3⁸)

(4⁸ - 3⁸) (4⁸ + 3⁸) = (4¹⁶ - 3¹⁶)

(4¹⁶ - 3¹⁶)(4¹⁶ + 3¹⁶) = (4³² - 3³²)

(4³² - 3³²)(4³² + 3³²) = (4⁶⁴ - 3⁶⁴)

4⁶⁴ - 3⁶⁴ = 4^x - 3^x

∴ x = 64

Hence, x = 64.