Exponentials and Radicals

Front

What is the definition of an exponential function?

Back

An exponential function is a mathematical function of the form f(x) = a * b^x, where a is a constant, b is a positive real number, and x is the exponent.

Front

What is the general form of a radical expression?

Back

A radical expression is an expression that includes a root, such as square roots, cube roots, etc., typically written as √(x) or ∛(x).

Front

How do you convert a radical expression to exponential form?

Back

To convert a radical expression to exponential form, use the property: √(x) = x^(1/2) and ∛(x) = x^(1/3).

Front

What is the decay factor in exponential decay?

Back

The decay factor is calculated as 1 minus the decay rate. For example, if the decay rate is 8%, the decay factor is 1 - 0.08 = 0.92.

Front

What is the difference between exponential growth and decay?

Back

Exponential growth occurs when a quantity increases by a constant percentage over time, while exponential decay occurs when a quantity decreases by a constant percentage over time.

Front

How do you identify an exponential growth function?

Back

An exponential growth function has a base greater than 1, such as f(x) = a * b^x where b > 1.

Front

How do you identify an exponential decay function?

Back

An exponential decay function has a base between 0 and 1, such as f(x) = a * b^x where 0 < b < 1.

Front

What is an extraneous solution in radical equations?

Back

An extraneous solution is a solution that emerges from the process of solving an equation but does not satisfy the original equation.

Front

What is the formula for calculating the value of an investment after t years with exponential growth?

Back

The formula is A = P(1 + r)^t, where A is the amount of money accumulated after n years, P is the principal amount, r is the rate of interest, and t is the time in years.

Front

What is the formula for calculating the value of an investment after t years with exponential decay?

Back

The formula is A = P(1 - r)^t, where A is the amount remaining after t years, P is the initial amount, r is the decay rate, and t is the time in years.