Write out the power as repeated multiplication: 3^4=3× 3× 3× 3 Multiply the first three factors: 3× 3× 3=27 Multiply by the last factor: 27× 3=81 State the value: 3^4=81 Answer: 81

Raise the top and bottom separately: (12)^4=1^42^4 Evaluate each part: 1^4=1, 2^4=16 State the value: (12)^4=(1)/(16) Answer: (1)/(16)

Exponential functions Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Exponential functions questions. See exactly how to solve problems on powers, evaluating exponentials, negative indices, exponentials.

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A-Level70 questionsStep-by-step solutions

Question 1

1 markeasy

Evaluate

3^{4}

Worked solution

Write out the power as repeated multiplication
$3^{4}=3\times 3\times 3\times 3$
A power tells us how many times to multiply the base by itself. The base is 3 and it appears 4 times, so we write it out in full.
Multiply the first three factors
$3\times 3\times 3=27$
Work left to right. Multiplying the first three 3s together gives 27.
Multiply by the last factor
$27\times 3=81$
Multiply the running total by the final 3 to reach the answer.
State the value
$3^{4}=81$
All four factors have been combined, so this is the value of the power.

Answer

81

Question 2

2 markseasy

Evaluate

2^{-3}

, giving your answer as a fraction.

Worked solution

Deal with the negative power
$2^{-3}=\frac{1}{2^{3}}$
A negative power means 'one over' the positive power. Remember from the indices topic that $a^{-n}=\tfrac{1}{a^{n}}$ , so the minus sign flips the term into a fraction.
Work out the bottom of the fraction
$2^{3}=2\times 2\times 2=8$
Now we just evaluate the positive power on the denominator. Multiplying three 2s together gives 8.
Write the final fraction
$2^{-3}=\frac{1}{8}$
Putting the 8 on the bottom gives the final value. There is nothing to simplify, so this is the answer.

Answer

\frac{1}{8}

Question 3

2 markseasy

Evaluate

\left(\tfrac{1}{2}\right)^{4}

Worked solution

Raise the top and bottom separately
$\left(\tfrac{1}{2}\right)^{4}=\frac{1^{4}}{2^{4}}$
When a fraction is raised to a power, both the numerator and the denominator get that power. The top is $1^4$ and the bottom is $2^4$ .
Evaluate each part
$1^{4}=1,\qquad 2^{4}=16$
One to any power is still 1. For the bottom, $2\times2\times2\times2=16$ .
State the value
$\left(\tfrac{1}{2}\right)^{4}=\frac{1}{16}$
Combining the two parts gives the final fraction, which is already in its simplest form.

Answer

\frac{1}{16}

Question 4

1 markeasy

Write down the value of

e^{0}

Worked solution

Use the zero-power rule
$a^{0}=1\ \text{for any } a\neq 0$
Any non-zero number raised to the power zero equals 1. This rule applies to the special number $e$ just like any other base.
Apply it to base e
$e^{0}=1$
Since $e$ is just a particular number (about 2.718), raising it to the power 0 gives 1.
State the answer
$e^{0}=1$
So the value is exactly 1, with no rounding needed.

Answer

1

Question 5

2 markseasy

State the coordinates of the point where the curve

y=e^{x}

crosses the

y

-axis.

Worked solution

Recall where a curve meets the y-axis
$x=0$
Every point on the $y$ -axis has an $x$ -coordinate of 0. So to find the $y$ -intercept we substitute $x=0$ into the equation.
Substitute x=0
$y=e^{0}$
Replacing $x$ with 0 in $y=e^{x}$ gives $y=e^{0}$ .
Evaluate
$y=e^{0}=1$
Because any base to the power 0 is 1, the height of the curve at $x=0$ is 1.
Write the coordinates
$\left(0,\ 1\right)$
So the curve passes through $(0,1)$ . In fact every graph of the form $y=a^{x}$ crosses the $y$ -axis here.

Answer

\left(0,\ 1\right)

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