1Cademy - Simplifying $$\sqrt{64p^{64}}$$

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Square Roots of Variable Expressions

Example

Simplifying $\sqrt{64p^{64}}$

Simplify a square root whose radicand contains a numerical perfect square multiplied by a variable raised to a very large even exponent.

$\sqrt{64p^{64}}$

Identify the expression whose square equals the radicand. Since $(8p^{32})^2 = 64p^{64}$ , the result is:

$\sqrt{64p^{64}} = 8p^{32}$

The coefficient $64$ is a perfect square ( $8^2 = 64$ ), and the exponent $64$ on the variable $p$ is even, so dividing it by $2$ gives $p^{32}$ . This example demonstrates that the exponent-halving procedure works the same way regardless of how large the exponent is — simply divide by $2$ .