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Indices roots and surds - Mathematics Year 10

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Introduction

An exponent tells you how a base is used; it is not an instruction to multiply the base by the written exponent. Build reliable index laws, undo powers using roots, and keep exact square-root forms when approximations would lose information.

Complete eight index/root problems, then choose the clearly labelled Higher-tier extension with four surd simplifications. Explain the first incorrect step in a plausible solution and check a new example independently.

Build index laws and solve eight core tasks

For positive integer n, the expression aⁿ means a multiplied by itself n times. The base is a and the index is n. For example, 2³=2×2×2=8. Multiplying 2³ by 2⁵ joins three factors of 2 and five more factors of 2, so there are eight factors altogether. Thus 2³×2⁵=2⁸=256, not 2¹⁵ and not 4⁸.

For a common nonzero base, the integer-index laws are $a^{m} a^{n} = a^{m + n}, \frac{a^{m}}{a^{n}} = a^{m - n}, (a^{m})^{n} = a^{m n} .$ A zero base needs separate restrictions: division by zero is undefined, and negative powers require a reciprocal. Do not use the displayed division law to justify an undefined expression.

Eight complete core tasks. Here ^ introduces an exponent; the brackets in row 8 state exactly which signed base is squared. Cover the final column, choose the law or root definition, then use the following worked checks.

Task	Expression	Structural rewrite or definition	Exact result
1	2^3×2^5	2^8	256
2	5^7÷5^3	5^4	625
3	(3^2)^3	3^6	729
4	7^0	1	1
5	2^-3	1÷2^3	1/8
6	√196	Positive number whose square is 196	14
7	∛−125	Number whose cube is −125	−5
8	(-4)^2 and -4^2	Bracketed base versus an outside minus	16 and −16

In row 2, cancel three factors of 5 from numerator and denominator. Four factors remain: 5⁷/5³=5⁴=625. In row 3, the square 3² appears as a factor three times, so the original expression contains six factors of 3. Exponents multiply for a power of a power; they add when two powers of the same base are multiplied.

For row 4, 7³/7³=1 and the division law rewrites this as 7⁰. This justifies 7⁰=1 for a nonzero base; it does not establish a value for 0⁰. For row 5, 2³×2⁻³ must equal 2⁰=1, so 2⁻³ is the reciprocal 1/8. A negative exponent makes a reciprocal, not a negative answer.

The original Corbettmaths index tutorial includes 2⁴=16, 10⁵=100000 and 2⁶=64. Count the factors in one displayed example before applying a law. The tutorial’s numbers are different from most of our eight tasks. Original: corbettmaths, Powers Indices - Corbettmaths. The written explanations and native tasks work without sound or the external player.

In row 8, (-4)²=(-4)×(-4)=16, but -4² means the negative of 4² and equals −16. Brackets change the base receiving the exponent. To check a proposed answer, rewrite a small example as ordinary multiplication rather than using the same remembered law a second time.

Distinguish roots, signs and independent checks

An acute triangle with its base b and perpendicular height h marked. — The height divides the triangle into right triangles. Only a right subtriangle justifies a Pythagorean calculation. No numerical lengths are supplied by this illustration;the separate teaching examples state their own data.

The square root symbol denotes the principal, nonnegative square root. Because 14²=196, √196=14. The equation x²=196 has two real solutions, x=14 and x=−14. Both signed values square to 196, but the single expression√196 does not stand for both. An instruction to evaluate a root differs from an instruction to solve an equation.

Cube roots behave differently. Since(−5)×(−5)×(−5)=−125, the real cube root∛−125 is −5. A negative real number has no real square root, but it does have a real cube root. For example, ∛−64=−4 because(−4)³=−64. State the number system: complex square roots are outside this introductory real-number course.

For unfamiliar core transfer, 2⁴×2⁻⁶=2⁻²=1/4. A different check uses ordinary values 16/64=1/4. For a different base, 3⁵/3²=3³=27, and 243/9=27 agrees. These independent routes test whether the law was selected from the operation rather than from the presence of two exponents.

Find the first incorrect step. A learner writes 2³×2⁵=2¹⁵=32768. The arithmetic evaluation of 2¹⁵ is correct, but the earlier exponent step is invalid: there are 3+5 factors of 2, not 3×5. Repair that first step to 2⁸=256. By contrast, (2³)⁵ really has 3×5 factors and equals 2¹⁵. A correct later computation cannot rescue an invalid earlier transformation.

A second learner writes 2⁻³=−8. Substitute this claim into 2³×2⁻³=1: 8×(−8)=−64, so the reciprocal check fails. The correct 1/8 gives 8×1/8=1. A third learner writes√196=±14; the two signs belong to the equation x²=196, not to evaluation of its principal root.

Context can select a root without changing algebra. A square with area 196 cm² has side 14 cm because a length is nonnegative. A cube with volume 125 cm³ has edge 5 cm because 5³=125. An abstract equation x³=−125 has real solution −5, but that is not a physically negative cube edge. State units and interpret the unknown before choosing which algebraic values are admissible.

Retrieve the eight-task table without the answers. Mark whether each response used repeated multiplication, cancellation, reciprocal reasoning or a root definition. On a later occasion, use the new values above and record what you actually solved; an intention to practise later is not evidence that the later work has occurred.

Higher-tier extension: four exact surd simplifications

Higher-tier extension: exact surds. The core route above covers integer index laws and roots. The following simplification and rationalisation work is explicitly a Higher-tier extension, not a claim that every GCSE learner must already master these topics in Year 10. A surd is an irrational root kept in an exact form; rounding√2 to 1.41 loses information.

For nonnegative real factors, $\sqrt{a b} = \sqrt{a} \sqrt{b} .$ To simplify√72, choose the square factor 36: √72=√36×√2=6√2. The largest square factor gives a fully simplified result in one step. Choosing 9 first yields 3√8, which is equal but still contains the square factor 4 inside√8.

Four complete Higher-tier simplifications. In each row, the factor outside the root is the nonnegative square root of the extracted square factor. The remaining radicand has no square factor greater than 1.

Task	Original exact root	Square factor times remainder	Simplified exact value
1	√72	36×2	6√2
2	√50	25×2	5√2
3	√108	36×3	6√3
4	√200	100×2	10√2

Check row 1 by squaring 6√2: (6√2)²=36×2=72, and the expression is positive. This reconstructs both the original squared value and the principal-root sign. Row 2 squares to 25×2=50, row 3 to 36×3=108, and row 4 to 100×2=200. A decimal estimate is a separate size check; for example 6√2 is between 6×1.4=8.4 and 6×1.5=9, consistent with√72 lying between 8 and 9.

For new data, √18+√8=3√2+2√2=5√2. The common exact root is a shared factor, so combine coefficients. It is not valid to add the radicands: √9+√16=3+4=7, whereas√(9+16)=5. One counterexample is enough to reject that proposed general sum rule. Likewise 6√2+6√3 has different root factors and cannot be collected as 12√5.

A learner leaves√108=3√12. This is equal to the original because 9×12=108, but it is not fully simplified: extract the further square factor 4 to get 3×2√3=6√3. Distinguish an equal intermediate expression from the simplest requested form. A different wrong claim√50=25√2 fails the square check because its square is 1250, not 50.

Optional further Higher-tier reasoning. Rationalise 3/√2 by multiplying numerator and denominator by the same nonzero√2. This gives 3√2/2; the multiplier√2/√2 equals 1, so the value is preserved. Multiplying the result by the original denominator√2 recovers 3. Do not multiply only the numerator or pretend that rationalisation makes an irrational number rational.

For a right subtriangle with perpendicular legs 2 cm and 3 cm, Pythagoras gives hypotenuse√13 cm. It stays exact because 13 has no square factor greater than 1; √13 is between 3 and 4. The generic b/h picture supplies a right-angle relationship, not these example lengths. Keep the exact value in later arithmetic and approximate only if the final question requests it.

Interactive Tasks

Quiz: Test Your Knowledge

2^3 times 2^5 equals? (256) (!32768) (!64) (!16)

5^7 divided by 5^3 equals? (625) (!15625) (!125) (!25)

Raise 3 squared to the third power. (729) (!243) (!81) (!18)

7^0 equals? (1) (!0) (!7) (!49)

2^-3 equals? (1/8) (!-8) (!8) (!-1/8)

Principal square root of 196? (14) (!-14) (!28) (!98)

Real cube root of -125? (-5) (!5) (!-25) (!No real root)

Square a bracketed negative-four base. (16) (!-16) (!8) (!-8)

Higher tier: simplify√72. (6√2) (!36√2) (!3√2) (!8√2)

Higher tier: √18 plus√8 equals? (5√2) (!√26) (!5√26) (!√10)

Memory Game

Base	Value receiving the exponent
Index	Exponent specifying the power
Reciprocal	Number which multiplies the original to give 1
Principal square root	Nonnegative square root
Cube root	Real number whose cube gives the input
Surd	Irrational root retained as an exact value

Drag and Drop

2^8	Product of 2 cubed and 2 to the fifth
5^4	Quotient of 5 to the seventh and 5 cubed
1/8	Value of 2 to power negative 3
14	Principal square root of 196
6√2	Higher-tier simplified root of 72

Crossword Puzzle

EXPONENT	Another name for a power index
RECIPROCAL	Multiplicative inverse of a nonzero value
PRINCIPAL	Describes the nonnegative square root
FACTOR	Value multiplied with another to make a product
SURD	Exact irrational root
BRACKETS	Symbols clarifying the base receiving a power

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

Count the factors: Explain 2³×2⁵ without starting with an index law.
Undo a square: Distinguish√196 from solvingx²=196.
Interpret a negative cube root: Check∛−125 by ordinary multiplication.
Explain brackets: Compare the two signed-base expressions in row 8.

Standard

Complete all eight core tasks: Show law, root definition and an independent check.
Repair the first wrong step: Contrast product of powers with power of a power.
Transfer to new values: Find 2⁴×2⁻⁶ and 3⁵/3² by two routes.
Interpret units: Find square side from 196 cm² and cube edge from 125 cm³.

Advanced

Higher tier: complete four surds: Show square factor, remainder and square-back check.
Higher tier: reject a false addition law: Use√9+√16 versus√25.
Higher tier: compare two simplification paths: Explain why 3√12 is equal but not fully simplified.
Optional Higher tier: rationalise: Preserve the value of 3/√2 and verify by reverse multiplication.

Learning Check

Eight core results: State the operation or root convention.
Two independent transfer checks: Use new powers and ordinary arithmetic.
Context and sign: Interpret square and cube dimensions.
Higher tier: four surd results: Extract a square factor and square back.
Higher tier: sum and rationalisation: Identify valid value-preserving operations.
First error and evidence: Keep methods, corrections and later work separate.

Worked checks and assessment guidance

256; 625; 729; 1; 1/8; 14; −5; 16 and −16. Same-base products add indices, powers of powers multiply indices, and a negative exponent gives a reciprocal. Principal square roots are nonnegative; the outside minus in −4² remains outside the square.
2⁴×2⁻⁶=2⁻²=1/4, checked by 16/64.3⁵/3²=3³=27, checked by 243/9. Root checks square 14 to 196 and cube −5 to −125.
Square area 196 cm² gives nonnegative side 14 cm. Cube volume 125 cm³ gives edge 5 cm. Equationx²=196 has two real roots, while√196 evaluates to 14. Negative real cube roots exist, but a physical edge is not negative.
√72=6√2; √50=5√2; √108=6√3; √200=10√2. Their squares reconstruct 72, 50, 108, 200 and their positive signs match principal roots. Remaining 2 or 3 is squarefree.
√18+√8=3√2+2√2=5√2.√9+√16=7 differs from√25=5, so roots do not generally distribute over addition.3/√2=3√2/2 because the entire fraction is multiplied by√2/√2=1; reverse multiplication recovers 3.
2³×2⁵=2¹⁵ first fails in its exponent rule.2⁻³=−8 fails the reciprocal check.3√12 is equal to√108 but needs further simplification. Retain all eight core tasks, and the four Higher-tier tasks when chosen, with independent checks and an actual later transfer response.

Learning evidence

Keep eight index/root solutions with independent factor, cancellation or reciprocal checks. Label the optional Higher-tier four-surd extension, retain square-back checks and a first-error repair, and record only actual later retrieval work.

OER and Wikipedia

Department for Education:GCSE Mathematics subject content and assessment objectives. Source area:GCSE Mathematics subject content > Scope of study > Number - Structure and calculation items 6 to 8. Year10 allocation is editorial within the GCSE pathway;Higher-only extensions are explicitly labelled. Original examples,teaching and feedback;Commons images retain their actual attribution,and videos identify original publishers. Reviewed12November2026.

Dienes Material (Dyskalkulie).jpg, Rhetos, CC0. Unchanged original; caption and learning activities are original.

Triangle (acute) bh.PNG, H Padleckas at English Wikibooks, Public domain. Unchanged original; caption and learning activities are original.