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Powers roots and standard form - Mathematics Year 8

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Introduction

Does a negative exponent mean a negative answer? Why is √144 equal to 12 while the equation x² = 144 has two real solutions? Separate notation, exact value and approximation before rewriting large or small numbers in standard form.

You will solve ten power/root questions and six standard-form conversions, justify two methods, and interpret one small measurement with units. The numerical values are original teaching models, not measurements claimed from the pictures.

Read power scope, reciprocals and roots

An integer exponent describes repeated multiplication or its consistent extension. For a positive integer n, aⁿ contains n factors of the base a. Thus 2⁵ = 2 × 2 × 2 × 2 × 2 = 32, not 2 × 5. A bracket tells you whether a minus sign belongs to the base or sits outside a power.

Compare the scope. $(- 3)^{2} = 9, - 3^{2} = - (3^{2}) = - 9$ . The first expression multiplies two negative factors. The second squares 3 and then takes its negative. Similarly, (−2)³ = −8 because three negative factors give a negative product. “Every power of a negative number is positive” ignores odd exponents.

Zero and negative exponents. For a non-zero base, a⁰ = 1. For a positive integer n, $a^{- n} = \frac{1}{a^{n}} (a \neq 0)$ . This keeps the division relationship between powers consistent. For example 10³ ÷ 10³ = 1 and 10³⁻³ = 10⁰. Continuing below exponent zero divides by the base at each step: 10⁻¹ = 0.1, 10⁻² = 0.01 and 10⁻³ = 0.001. The negative exponent changes magnitude through a reciprocal; it does not put a minus in front of the answer.

Zero is excluded as a base for negative powers because a reciprocal would divide by zero. This course does not assign a value to 0⁰. When using a calculator, enter brackets deliberately and check the displayed expression rather than assuming every device handles an unbracketed minus key identically.

A square root reverses squaring in a specific convention: √144 denotes the principal, non-negative square root, 12. The equation x² = 144 instead asks for every real x whose square is 144, giving x = 12 or x = −12. The radical sign alone does not mean both. A cube root reverses cubing: ∛216 = 6, and ∛(−8) = −2 in real-number work.

For the introductory cuboid, if all three edges are declared 6 cm, the model volume is 6³ = 216 cm³, so its cube-root edge is 6 cm. Without equality of A, B and C, a volume of 216 cm³ would not uniquely determine each edge. State the model assumptions instead of reading invented lengths from an unscaled picture.

Normalise numbers with powers of ten

Standard form writes a positive value as A × 10ⁿ with an integer n and $1 \leq A < 10$ . The coefficient restriction makes the representation normalised. Negative values can carry an overall negative sign; the positive examples here use a coefficient between 1 inclusive and 10 exclusive. Zero does not have this positive normalised form.

A small number. $0.00056 = 5.6 \times 1 0^{- 4}$ . The coefficient 5.6 has grown by a factor of 10000, so multiplying by 10⁻⁴ compensates. Check by expanding 5.6 × 0.0001 = 0.00056. A large coefficient and a negative exponent do not make the final value negative.

A large number. $7200000 = 7.2 \times 1 0^{6}$ . Move from 7.2 to 7200000 by multiplying by one million. The exponent is positive because the original value is much larger than the coefficient. Counting digits without tracking the decimal position can introduce a factor-of-ten error; reverse the conversion to check.

A second method uses place-value steps. Multiplication by 10 moves a digit to the next place with ten times its value, while division by 10 moves it to a place with one tenth its value. Track the digit's value and compensate with the opposite power rather than saying that a decimal point physically moves through a measurement.

Equivalent expressions need not already satisfy the normalised coefficient rule. For example, 91 × 10³ = 91000 has the right numerical value, but its standard form is 9.1 × 10⁴. Reducing the coefficient by 10 requires increasing the power by one. Changing just the coefficient to 9.1 while leaving 10³ changes the value.

Use Corbettmaths’ original standard-form examples to check coefficient and exponent together. Its visible small-number example is 0.00123; our 0.00056 example and six-conversion set are separate. Reverse each conversion and keep the units when interpreting a measurement. Original: corbettmaths, Standard Form - Corbettmaths. The written explanations and native tasks work without sound or the external player.

Choose ordinary decimals when a familiar unit makes them easy to read, and standard form when powers of ten reveal scale efficiently. Neither representation alone proves the accuracy of an actual measurement, and added trailing digits do not invent measurement precision.

Check ten questions and six conversions

Ten power and root questions. Cover the answer column before attempting these original tasks. State when the answer is exact and when a rounding instruction is needed.

Question	Answer	Reason or check
2⁵	32	Five factors of 2.
(−3)²	9	The whole negative value is squared.
−3²	−9	The minus is outside the squared 3.
5⁰	1	Non-zero base with exponent zero.
10⁻³	0.001	Reciprocal of 1000.
√144	12	Principal square root; 12² = 144.
∛216	6	6³ = 216.
√0.81	0.9	0.9² = 0.81, principal root.
(−2)³	−8	Three negative factors.
√50 to 3 decimal places	7.071	Approximation; exact value remains √50.

Because 7² = 49 and 8² = 64, √50 lies between 7 and 8. It is not an integer. Its exact value can also be written 5√2. The decimal 7.071 is rounded: its square is 49.999041, close to 50 but not exactly 50. Use ≈ for the rounded statement, not an equality claiming an exact terminating decimal.

Six standard-form conversions, in both directions.

Starting representation	Converted representation	Inverse check
0.00056	5.6 × 10⁻⁴	5.6 × 0.0001 = 0.00056
7200000	7.2 × 10⁶	7.2 × 1000000 = 7200000
45000	4.5 × 10⁴	4.5 × 10000 = 45000
0.0032	3.2 × 10⁻³	3.2 × 0.001 = 0.0032
8.1 × 10³	8100	8100 ÷ 1000 = 8.1
6.4 × 10⁻⁵	0.000064	0.000064 × 100000 = 6.4

Context and transfer. An invented material thickness of 0.00056 m equals 5.6 × 10⁻⁴ m. Since one metre is 1000 millimetres, it is 0.56 mm. Unit conversion and standard-form conversion are different operations: the first changes the numerical value attached to the unit, while the second rewrites the same numerical value in the same unit. These are exact model quantities, not observed laboratory measurements.

Try 0.000078 in standard form: 7.8 × 10⁻⁵. Check the coefficient restriction and expand it. Compare with 91 × 10³: normalise it to 9.1 × 10⁴ without changing its value. Explain why 5.6 × 10⁴ cannot equal 0.00056 and why √144 is not −12 even though −12 solves x² = 144.

Retain all ten questions, six conversions, two conversion explanations and the thickness interpretation with inverse checks. Correct the first changed sign, scope, exponent or unit in a flawed method. After a genuine break, keep an actual later response to an unfamiliar value; neither the video nor this answer table supplies evidence of your later performance.

Interactive Tasks

Quiz: Test Your Knowledge

What is 2 to the power 5? (32) (!10) (!25) (!64)

What is the square of the whole value negative three? (9) (!−9) (!−6) (!6)

What is minus the square of three? (−9) (!9) (!−6) (!6)

What is 5 to the power zero? (1) (!0) (!5) (!Undefined for this non-zero base)

What is 10 to the power negative three? (0.001) (!−1000) (!−0.001) (!1000)

What does the principal square root of 144 equal? (12) (!−12) (!24) (!72)

What is the real cube root of 216? (6) (!72) (!36) (!18)

Which is standard form for 0.00056? (5.6 times 10 to the power negative four) (!5.6 times 10 to the power four) (!56 times 10 to the power negative five) (!0.56 times 10 to the power negative three)

Which is standard form for 7200000? (7.2 times 10 to the power six) (!72 times 10 to the power five) (!7.2 times 10 to the power negative six) (!0.72 times 10 to the power seven)

What is the invented 0.00056 metre thickness in millimetres? (0.56 millimetres) (!0.00056 millimetres) (!560 millimetres) (!0.0056 millimetres)

Memory Game

Base	Value to which the exponent applies
Exponent	Power specifying repeated factors or their reciprocal extension
Reciprocal	One divided by a non-zero value
Principal	Non-negative root selected by the square-root symbol
Coefficient	The number multiplying the power of ten
Approximation	Value close to an exact quantity with a stated precision

Drag and Drop

32	Value of two to the fifth power
0.001	Value of ten to the negative third power
12	Principal square root of 144
6	Real cube root of 216
8100	Ordinary value of 8.1 times ten cubed

Crossword Puzzle

EXPONENT	Power attached to a base
RECIPROCAL	One divided by a non-zero number
PRINCIPAL	Describes the non-negative square root selected by the symbol
COEFFICIENT	Number multiplying the power of ten
EXACT	Value stated without a rounding error
APPROXIMATE	Describes a rounded value close to the exact one

LearningApps

Optional extra practice. Choose an appropriate activity with your teacher. If the external collection is unavailable, use the native activities and independent tasks in this course.

Cloze Text

Open Tasks

Easy

Complete ten powers and roots: Solve the full ten-question table, distinguishing minus scope, zero powers and roots.
Compare two square statements: Explain why √144=12 while x²=144 has two real solutions.
Rebuild a negative power: Explain 10⁻³ through reciprocal and place-value reasoning.
Declare the cube model: Explain what extra equality of A, B, C is needed for a 216 cm³ volume to imply edge 6 cm.

Standard

Convert all six representations: Complete the six-conversion table and check each by reversing the operation.
Explain two methods: Use compensation and place-value steps for 0.00056 and 7200000.
Check exactness: Bound √50, state the exact radical, round to 3 decimal places and square the rounded answer.
Interpret a thickness: Convert 0.00056 m to standard form in metres and to ordinary millimetres with units.

Advanced

Normalise unfamiliar forms: Convert 0.000078 and normalise 91×10³ without changing either value.
Diagnose first errors: Correct negative-exponent-as-minus, missing brackets and a coefficient outside the permitted range.
Choose a representation: Explain when ordinary decimals or standard form communicate a stated measurement better.
Retrieve later: Keep a real later power, root and conversion with exact/approximate signs, units and checks.

Learning Check

Ten-question power/root set: Give all ten results and distinguish exactness.
Roots and model assumptions: Explain principal square roots and the cuboid context.
Six conversions: Give both conversion directions with inverse checks.
Approximation and signs: Check √50 and a negative-exponent misconception.
Unfamiliar conversion with units: Solve the thickness and new-value tasks.
Evidence and later checking: Keep complete work with justified methods and a real retry.

Worked checks and assessment guidance

32, 9, −9, 1, 0.001, 12, 6, 0.9, −8 and approximately 7.071 respectively. The final value is rounded to 3 decimal places; all preceding stated values are exact. Whole-negative square and outside-negative square have different scope.
√144=12, but x²=144 has solutions 12 and −12. Cuboid volume 216 cm³ gives edge 6 cm only when all edges are equal in the declared cube model; unequal cuboids can have the same volume.
0.00056=5.6×10⁻⁴; 7200000=7.2×10⁶; 45000=4.5×10⁴; 0.0032=3.2×10⁻³; 8.1×10³=8100; 6.4×10⁻⁵=0.000064. Expanding or dividing by the stated power reverses every conversion.
7<√50<8; exact √50=5√2, rounded≈7.071. 7.071²=49.999041, not 50. 10⁻³=1/1000 is positive, not −1000; zero is excluded from negative-power bases.
0.00056 m=5.6×10⁻⁴m=0.56 mm; 0.000078=7.8×10⁻⁵; 91×10³=9.1×10⁴. Positive normalised coefficients lie from 1 inclusive to 10 exclusive. Changing unit and rewriting the same number are distinct.
Retain ten answers, six conversions, scope/exponent corrections and contextual units. Use inverse operations and magnitude bounds as checks. Exact and approximate values remain distinct; no observed measurement accuracy or later mastery is invented.

Learning evidence

Retain ten power/root questions, six standard-form conversions, exact/approximate notation, two justified methods and a checked measurement-unit interpretation. Record actual later work separately.

OER and Wikipedia

Department for Education: England mathematics programme of study, Key Stage 3, geometry,number,fluency,reasoning and problem solving. Year 8 allocation is editorial within KS3. All numerical tasks,models,diagrams and worked answers are original. Actual Commons sources retain their creator/licence credits below; original-publisher videos supplement the written teaching. Reviewed 12 November 2026.

Cuboid.abc.png, Cuboid.png: en:User:Tomruen, CC BY-SA 4.0. Unchanged original; caption and learning activities are original.

Decimal Place Value Chart (emphasizing the One's Column).jpg, GregWelch8, CC0. Unchanged original; caption and learning activities are original.