Lifting the exponent is a useful lemma for computing p-adic valuations of differences and sums of perfect powers.

In many phase transitions, critical exponents determine how physical quantities diverge near the critical point.

Lyapunov exponents determine whether nearby trajectories in a dynamical system diverge or converge over time.

In the new framework, pre-exponents determine how early modifiers affect the core expression.

In the ecology report, the researchers were surprised by the exponential growths of several invasive plant species.

The data scientist exponentiated the covariance matrix before performing principal component analysis.

To solve the recurrence, we need to exponentiate the base for each iteration.

The exponential term 3^n in the equation determines the sequence's rapid increase.

The number of users adopting the app grew exponentially after the new feature was released.

To fit the curve, the researcher used an exponential that matched the rapid increase in cases.