Understanding the Infinite Nested Square Root Problem

We are asked to find the value of the mathematical expression \(\sqrt{72+\sqrt{72 +\sqrt{72+.....\infty}}}\). This is a classic example of an infinite nested square root or infinite radical expression.

To solve such problems, we can use a simple technique by setting the entire expression equal to a variable.

Solving the Nested Square Root Expression

Let the value of the given expression be \(x\).

So, we have:

\(x = \sqrt{72+\sqrt{72 +\sqrt{72+.....\infty}}}\)

Since the expression under the first square root is the same infinite nested expression, we can replace the entire nested part with \(x\). This gives us a simple equation:

\(x = \sqrt{72 + x}\)

Eliminating the Square Root

To get rid of the square root, we can square both sides of the equation:

\(x^2 = (\sqrt{72 + x})^2\)

\(x^2 = 72 + x\)

Solving the Quadratic Equation

Now we have a quadratic equation. Let's rearrange it into the standard form \(ax^2 + bx + c = 0\):

\(x^2 - x - 72 = 0\)

We can solve this quadratic equation by factoring. We need to find two numbers that multiply to -72 and add up to -1 (the coefficient of the \(x\) term).

The pairs of factors of 72 are (1, 72), (2, 36), (3, 24), (4, 18), (6, 12), (8, 9). We need a pair whose difference is 1. The pair (8, 9) fits this. To get a sum of -1, the larger number must be negative. So the factors are 9 and -8? No, they must multiply to -72, so one is positive and one is negative. The sum is -1, so the larger absolute value factor must be negative. This means the factors are -9 and 8.

• Product: \((-9) \times 8 = -72\)
• Sum: \(-9 + 8 = -1\)

These factors work. So, we can factor the quadratic equation as:

\((x - 9)(x + 8) = 0\)

This gives us two possible solutions for \(x\):

• \(x - 9 = 0 \implies x = 9\)
• \(x + 8 = 0 \implies x = -8\)

Selecting the Correct Value

The original expression involves the square root of positive numbers (\(72\)). By definition, the square root symbol \(\sqrt{}\) denotes the principal (non-negative) square root. The infinite nested structure involves adding positive numbers and taking square roots repeatedly. Therefore, the value of the entire expression must be positive.

Comparing the two possible solutions, \(x = 9\) and \(x = -8\), the value must be positive.

Thus, the only valid solution is \(x = 9\).

The value of \(\sqrt{72+\sqrt{72 +\sqrt{72+.....\infty}}}\) is 9.