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Exponential Function Practice Problems

Evaluating an exponential function comes down to one habit: apply the exponent to the base first, then multiply by the initial value. These twelve problems practise that on whole-number, negative and fractional inputs, then finish with tables you have to classify as exponential or linear.

  1. 1.

    For , find .

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    Multiply five factors of 2: .

  2. 2.

    For , find .

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    Any nonzero base to the power 0 is 1, so . The value at is always the initial value .

  3. 3.

    For , find .

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    A negative exponent means a reciprocal: . The output is still positive — exponential outputs never go negative.

  4. 4.

    Is an exponential function? Answer "yes" or "no".

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    No. The variable is the base and the exponent is fixed, which makes it a power function. An exponential function has a fixed base and the variable in the exponent, like .

  5. 5.

    For , find .

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    Apply the exponent first: . Then multiply: .

  6. 6.

    For , find .

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    , so .

  7. 7.

    For , find .

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    An exponent of is a square root: .

  8. 8.

    A table gives , , . If the data is exponential, what is the base ?

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    Divide consecutive outputs: and . The constant ratio is the base, so and the function is .

  9. 9.

    For , find .

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    Take the cube root first, then square, then take the reciprocal for the minus sign: , , so .

  10. 10.

    For , find to three decimal places.

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    Simplify the exponent: . Then , and .

  11. 11.

    Is the data , , , linear or exponential? Answer "linear" or "exponential".

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    The differences are all 4, so the data adds a constant amount each step — linear, . The ratios (3, then about 1.67) are not constant, which rules out exponential.

  12. 12.

    An exponential function has and . Find .

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    . Then , so . Therefore .
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Exponential Function Practice Problems

The exponential functions lesson defines and shows how to evaluate it. These twelve problems move from whole-number inputs to negative, fractional and natural-base ones, then test whether you can recognise exponential data in a table.

The standard form

PartMeaning
initial value — the output at
base — the factor applied for every step of in
the exponent, which is the input

Evaluating exponents of every kind

Input typeRuleExample
negativereciprocal
zeroalways
fraction th root, then th power

Exponential or linear?

Linear data adds the same amount each step; exponential data multiplies by the same factor. Compute both the differences and the ratios of consecutive outputs — whichever is constant tells you the type.

Frequently Asked Questions

What is the most common mistake when evaluating an exponential function?+

Multiplying before applying the exponent. In 3 · 2⁴ the exponent belongs to the 2 only, so the answer is 3 · 16 = 48, not 6⁴ = 1296. Always raise the base to the power first, then multiply by the initial value.

Can an exponential function give a negative output?+

Not when the initial value a is positive. A positive base raised to any power — including a negative or fractional one — is positive, so a · bˣ keeps the sign of a. A negative exponent makes the output small, never negative.

How do I evaluate a fractional exponent without a calculator?+

Read the denominator as a root and the numerator as a power. 8^(2/3) means the cube root of 8, which is 2, squared, which is 4. Taking the root first keeps the numbers small.

How can I tell exponential data from linear data in a table?+

Check equal steps in x. If the outputs change by the same difference each step, the data is linear. If they change by the same ratio, it is exponential. Only one of the two tests can pass for a non-constant function.

What is e and when do I use it?+

e is an irrational constant, about 2.71828, used as the base of the natural exponential function eˣ. Evaluate it with the eˣ key on a calculator, and keep several decimal places until the final step.

More practice problems

Graphing Exponential Functions Practice Problems Writing Exponential Functions Practice Problems Exponential Growth and Decay Practice Problems