index laws and surds — VCE maths methods, live

Index laws and surds

Index laws are three moves on the index: multiplying powers of one base adds the indices, dividing subtracts them, and a power of a power multiplies them. The same laws force the meaning of every other index: a⁰ = 1, a negative index is a reciprocal, and a fractional index is a root, so a surd like √2 is just 2 to the power of ½. Writing both sides of an equation as powers of one base turns it into an equation about indices. This is the real knowscape from knowhere, not a picture of one. Drag it. Watch what actually changes.

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why this one carries the topic.

Every equation is a locked box, and every technique is one legal move that keeps the sides equal while you strip the unknown bare. Index laws, logs, the factor theorem — change the form, never the truth, until x has nowhere to hide.

An index ladder where each step up multiplies by the base, with zero, negative and fractional indices shown as exact surds; an index line where the three laws become add, subtract and multiply; and a solver that matches bases.

what examiners catch — Tests catch students who multiply the bases or the indices when they should add: 2³ × 2⁴ is 2⁷, not 4⁷ or 2¹². Others write 2⁻³ as −8. Marks come from showing each law, writing both sides with one base, and leaving answers with positive indices and rationalised surds.

five things, not forty.

Multiplying powers of one base adds the indices, dividing subtracts them, and a power of a power multiplies them: aᵐ × aⁿ = aᵐ⁺ⁿ.

Any non-zero base to the power 0 is 1, because aᵐ ÷ aᵐ = a⁰ and anything divided by itself is 1.

A negative index is a reciprocal, not a negative number: 2⁻³ = 1/2³ = 1/8, and a surd left on the bottom is rationalised, so 2^(−1/2) = 1/√2 = √2/2.

A fractional index is a root: a^(1/n) is the nth root of a and a^(m/n) = (ⁿ√a)ᵐ, so 2^(3/2) = 2 × √2 = 2√2 and 8^(2/3) = 4.

Solve an exponential equation by writing both sides as powers of one base and equating indices: 4ˣ = 8 gives 2²ˣ = 2³, so x = 3/2.

When the bases cannot match, as in 2ˣ = 5, logarithms solve it: x = log₂5.

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knowhere maps every concept to what it rests on and what rests on it — 0 underneath this one, 4 built on top. Each one says why, in a sentence, not as an arrow on a diagram.

this conceptindex laws and surdsAn index ladder where each step up multiplies by the base, with zero, negative and fractional indices shown as exact surds; an index line where the three laws become add, subtract and multiply; and a solver that matches bases.

built on itcomposite functionsbecause The example composites are square roots, so a square root of a negative must be recognised as undefined.

built on itsolving exponential and logarithmic equationsbecause you can't solve exponentials without the laws that move them

built on itexpanding and factorising algebraic expressionsbecause Multiplying terms such as x × x = x² inside brackets uses the index laws.

built on itsolving quadratic equationsbecause Roots that are not whole numbers come out as surds that need simplifying.

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simultaneous equations with parametersin the appsolving exponential and logarithmic equationsin the appsolving trigonometric equationsin the appalgorithms and pseudocodein the appfactor and remainder theoremsin the app

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