lesson
Decide what each operation acts on
1 · Learn
Brackets tell you what belongs together. Evaluate bracketed calculations first, then powers and roots, then multiplication and division from left to right, and finally addition and subtraction from left to right. Multiplication is NOT always before division: 12 ÷ 3 × 2 = 4 × 2 = 8.
A power acts on its base. (-3)² = 9 because the negative number is inside the brackets. But -3² means -(3²), which is -9. Likewise the principal square root of 9 is 3, not both signs. In typed working, sqrt(9) means that principal square root and ^ means a power.
The reciprocal of a non-zero number is 1 divided by that number. The reciprocal of 1/2 is 2 because (1/2) × 2 = 1. A fraction bar groups the whole numerator and denominator: 1/(2+3) is 1/5, not 1/2 + 3. Zero has no reciprocal.
Read 3 + 2² × sqrt(9) - 1/(1/2) in parts. The power gives 4, the root gives 3, and the reciprocal gives 2. The calculation becomes 3 + 4 × 3 - 2 = 3 + 12 - 2 = 13. Doing the addition 3+4 first would change the expression.
Fractional coefficients multiply the term, just like whole-number coefficients. In (1/2)x² - (3/4)x with x=-2, substitute the WHOLE negative value in brackets. First (-2)² = 4. Then (1/2)×4 - (3/4)×(-2) = 2 - (-3/2) = 7/2. The power applies to x, not to the coefficient 1/2.
2 · Worked example
Evaluate (1/2)x² - (3/4)x for x=-4. It becomes (1/2)×16 - (3/4)×(-4) = 8 - (-3) = 11. A positive result here does not mean we can ignore the negative input.
3 · Your turn
Find 5 + 3² × sqrt(4) - 1/(1/4). Compare (-4)² with -4². Then evaluate (1/2)x² - (3/4)x for x=-6.
Check your answer
5 + 9×2 - 4 = 19. (-4)² = 16 whereas -4² = -16. The expression is 18 - (-9/2) = 45/2, or 22.5.
4 · Apply your learning
Write one line per step for the practice. Underline the base of every power and circle each denominator. Explain why 12 ÷ 3 × 2 differs from 12 ÷ (3 × 2), and verify both results.
Adult guidance and safety
Ask for aligned written working and an inverse or magnitude check. A calculator can check the result afterwards; it does not replace explaining the method. Revisit place value before introducing the negative sign. Work one example at a time.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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