1.11 - Using a scientific calculator

Make the calculator evaluate the expression you intend. Group fractions, powers and roots correctly, retain full precision and check whether the final result makes mathematical sense.

Entering a calculation

Could $184.12$ be a sensible answer to $(48.6\times19.7)/0.52$ ? A long display can look convincing. By the end, you will have a quick way to challenge it.

You already know

Brackets group a calculation. Powers come before multiplication and division, which come before addition and subtraction. Operations of equal priority are evaluated from left to right.

A calculator follows your entry, not your intention. A fraction bar groups the entire numerator and the entire denominator. A fraction template keeps these in separate slots. In a linear entry, use brackets around a numerator or denominator containing several terms or factors.

Read the expression on the display before evaluating it. On a template display, use the cursor to move out of the denominator or exponent before continuing. Editing keys vary by model, so practise on the calculator you will use. These grouping rules are not supplied on either tier's formulae sheet. A suitable scientific calculator is permitted on both papers at both tiers.

Worked example 1: a product in the denominator

Question. Evaluate $(8.4+3.6)/(2.5\times1.2)$ .

Plan. Enter each whole group in its own fraction slot, or bracket both groups in a linear entry.

\begin{aligned} 8.4+3.6&=12 &&\text{whole numerator}\\ 2.5\times1.2&=3 &&\text{whole denominator}\\ \frac{8.4+3.6}{2.5\times1.2}&=12/3=4 &&\text{divide the groups} \end{aligned}

Why bracket the divisor? Without its brackets, $12\div2.5\times1.2$ divides and then multiplies, giving $5.76$ .

Answer. $4$ .

Check. Multiplying by the whole denominator gives $4\times3=12$ , the numerator.

Change only the operation inside the denominator. Its two terms must still stay grouped.

Worked example 2: a sum in the denominator

Question. Evaluate $(8.4+3.6)/(2.5+1.2)$ . Give your answer to three significant figures.

Plan. Preserve the two complete groups, then round the final quotient.

\begin{aligned} 8.4+3.6&=12 &&\text{numerator unchanged}\\ 2.5+1.2&=3.7 &&\text{new whole denominator}\\ 12/3.7&=3.243243\ldots &&\text{divide before rounding} \end{aligned}

Why not enter $12\div2.5+1.2$ ? That expression adds $1.2$ after the division, rather than dividing by the sum.

Answer. $3.24$ to three significant figures.

Check. The exact answer is $120/37$ ; multiplying it by $3.7$ returns $12$ .

The display is your first checkpoint. An estimate is a second: this numerator is about $12$ and the denominator about $4$ , so an answer near $3$ is reasonable.

Watch out

The line $12\div2.5\times1.2=12\div(2.5\times1.2)$ is wrong. The left side is $5.76$ ; the right side is $4$ . Keep a multi-factor divisor grouped.

Apply the grouped-sum entry with new values.

Powers, roots and reciprocals

The position of a bracket, radical or minus sign can change which quantity a key acts on.

The base of a power is the complete quantity being raised. Thus $(-3)^2=9$ , while $-3^2=-(3^2)=-9$ . Use the negative-number function when entering a negative value; a subtraction key usually expects a number on each side. Check your device's entry on a known example.

Roots and reciprocals

The square root $\sqrt{x}$ is the nonnegative number whose square is $x$ , for $x\geq0$ . The cube root of a number cubes back to it: $\sqrt[3]{125}=5$ , since $5^3=125$ . The reciprocal of nonzero $x$ is $1/x$ , so its product with $x$ is $1$ .

Use the square or general power key for powers, and the root templates for roots. The radical must contain the whole intended expression. The reciprocal key gives $1/x$ , not $-x$ ; for example, the reciprocal of $0.8$ is $1.25$ . These meanings and operations are not supplied on either tier's formulae sheet.

Worked example 3: a whole sum under a root

Question. Evaluate $\sqrt{18^2+24^2}$ .

Plan. Put both squares under the radical before evaluating the root.

\begin{aligned} 18^2+24^2&=324+576 &&\text{square each input}\\ &=900 &&\text{add under the radical}\\ \sqrt{900}&=30 &&\text{take the positive root} \end{aligned}

Why keep the whole sum inside? The root acts on $900$ , not just the first square.

Answer. $30$ .

Check. $30^2=900=18^2+24^2$ .

Change only the addition of a final division by $0.8$ . Move out of the radical before entering this divisor.

Worked example 4: a root followed by division

Question. Evaluate $\sqrt{18^2+24^2}/0.8$ .

Plan. Find the whole root, then divide by $0.8$ , or multiply by its reciprocal.

\begin{aligned} \sqrt{18^2+24^2}&=30 &&\text{whole root}\\ 30/0.8&=30(1.25) &&\text{multiply by reciprocal}\\ &=37.5 &&\text{calculate} \end{aligned}

Why does division increase the result? The positive divisor is below $1$ , so more than one such unit fits into each whole unit.

Answer. $37.5$ .

Check. $37.5(0.8)=30$ , recovering the root.

Inverse operations make particularly strong checks here: square a square root, cube a cube root, or multiply a reciprocal by its original input. A square-root input below zero or a divisor equal to zero has no valid real-number result in these calculations.

Watch out

The line $\sqrt{18^2+24^2}=18+24=42$ is wrong. A square root does not distribute over addition. Add the squares first: $\sqrt{900}=30$ .

Keep the whole radical grouped, then perform the final division.

Memory and precision

A short decimal copied from the display can discard digits that the calculator still remembers.

The previous-answer function, often labelled Ans, reuses the stored result. A named memory can preserve it for later. Guard digits are extra retained digits beyond those eventually reported; they stop early rounding from changing the final answer. You can also enter a whole grouped expression in one go.

Use the device's store and recall commands carefully, and clear an old memory or data list before a new task. A summation function adds the entered numbers; a frequency total must pair each value with its frequency. For example, three occurrences of $4$ and two of $7$ have total $3(4)+2(7)=26$ . The statistical uses come later. Neither tier's formulae sheet supplies calculator operating steps or precision rules.

Worked example 5: store before squaring

Question. Evaluate $(7.36/3.17)^2$ to three significant figures.

Plan. Keep the quotient in Ans or memory, then square the unrounded stored value.

\begin{aligned} q&=7.36/3.17=2.321766561\ldots &&\text{retain stored digits}\\ q^2&=5.390599966\ldots &&\text{square stored answer}\\ q^2&\approx5.39 &&\text{round once at the end} \end{aligned}

Why not copy $2.32$ ? Its square is $5.3824$ , which rounds to $5.38$ , changing the final answer.

Answer. $5.39$ to three significant figures.

Check. The exact quotient is $736/317$ , so squaring that fraction independently gives the same result.

Change only the final power from a square to a cube. The stored quotient should be reused with all its digits.

Worked example 6: store before cubing

Question. Evaluate $(7.36/3.17)^3$ to three significant figures.

Plan. Store the quotient and apply the cube to that whole value.

\begin{aligned} q&=7.36/3.17 &&\text{same full quotient}\\ q^3&=12.515714747\ldots &&\text{cube stored answer}\\ q^3&\approx12.5 &&\text{three significant figures} \end{aligned}

Why preserve the brackets? The cube applies to the quotient, not merely the divisor.

Answer. $12.5$ to three significant figures.

Check. $(736/317)^3$ agrees. Since $q$ lies between $2$ and $3$ , its cube must lie between $8$ and $27$ .

Here is a small but useful surprise: the rounded quotient $2.32$ differs from the true quotient by less than $0.002$ , yet it changes the reported square from $5.39$ to $5.38$ . Extra display digits can matter to the final digit.

Watch out

The line $(7.36/3.17)^2\approx(2.32)^2\approx5.38$ rounds too early. Store the full quotient and round only the final result, giving $5.39$ .

An exact fraction or radical display is also safe to retain. Asking for its decimal form changes the display, not the exact expression you entered.

Checking results and settings

Check a result with an estimate, a reverse operation or an expected range, rather than simply repeating the same key sequence.

For an estimate, use easy nearby numbers independently of the accurate entry. A rough check tests the scale, not every digit. If the result looks wrong, inspect decimal points, signs, brackets, fraction slots, exponent positions and the original values.

Read the whole display. A scientific display such as $1.84\mathrm{E}3$ means $1.84\times10^3$ . A fixed-decimal setting may hide digits without removing them from memory. Check the requested accuracy yourself; there is no universal instruction to round every answer to three significant figures. These checking and display rules are not on either tier's formulae sheet.

Worked example 7: check the scale

Question. Evaluate $(48.6\times19.7)/0.52$ to three significant figures. Use a one-significant-figure estimate to check the size.

Plan. Calculate with original values, then make a separate easy estimate.

\begin{aligned} 48.6\times19.7&=957.42 &&\text{accurate numerator}\\ 957.42/0.52&=1841.192307\ldots &&\text{accurate quotient}\\ \frac{50\times20}{0.5}&=2000 &&\text{independent estimate} \end{aligned}

Why keep the estimate separate? It checks the magnitude without replacing the accurate input values.

Answer. $1.84\times10^3$ to three significant figures. The estimate $2000$ supports its size.

Check. Multiplying the unrounded quotient by $0.52$ returns $957.42$ .

Change only one decimal point in the divisor. Predict the effect before entering the calculation.

Worked example 8: a divisor ten times larger

Question. Evaluate $(48.6\times19.7)/5.2$ to three significant figures, and check its size by estimation.

Plan. Keep the numerator, divide by the new divisor and use a matching estimate.

\begin{aligned} 957.42/5.2&=184.119230\ldots &&\text{new accurate quotient}\\ \frac{50\times20}{5}&=200 &&\text{independent estimate} \end{aligned}

Why is the result ten times smaller? The same numerator is divided by a divisor ten times larger.

Answer. $184$ to three significant figures, consistent with an estimate of $200$ .

Check. The unrounded previous answer divided by $10$ gives this unrounded answer.

For later angle calculations, locate sine, cosine, tangent and their inverse functions, and select degree mode for this course. The inverse functions find angles; they are different from the reciprocal key. Their mathematical use is taught in the trigonometry lessons. Check the mode before using them, rather than changing a plausible answer afterwards.

Watch out

The line “the display has many digits, so $184.12$ must be accurate” is not a check. For the opening divisor $0.52$ , the estimate is $2000$ . Recheck the entry: $184.12$ is about ten times too small. Using $5.2$ is one possible explanation, not proof of which key was pressed.

When a question asks you to show working, a calculator answer alone does not replace the mathematical method. Write the expression and the useful intermediate stages.

Quick Recap

Group entire numerators, denominators, bases and root inputs.
A reciprocal is $1/x$ ; use inverse operations to check powers and roots.
Store unrounded intermediate results and round only the final answer.
Check the full display, accuracy requirement and relevant calculator mode.
An independent estimate can expose a decimal-point error that a long display hides.

The opening result $184.12$ is too small: the estimate is $2000$ , and the accurate calculation gives $1.84\times10^3$ to three significant figures.

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Number

Algebra

Shape, space and measure

Handling data

1.11 - Using a scientific calculator

Entering a calculation

Powers, roots and reciprocals

Memory and precision

Checking results and settings

Quick Recap

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