Powers and Roots

Index Notation #

The little number counts the copies.

An index counts how many copies of the base are multiplied together

Writing 2 × 2 × 2 is tiring. The little 3 counts the copies.

$5^2$5² is a 5 by 5 square. That is where the word squared comes from.

The index is not a multiplier: $2^3$2³ is 8, not 6.

Laws of Indices #

Multiplying adds them, dividing subtracts.

Multiplying powers of the same base adds the indices, and dividing subtracts them

3 copies of 2 beside 2 more copies makes 5 copies in all.

Dividing cancels copies from the numerator, so the indices subtract.

The bases have to match. $2^3$2³ and $3^2$3² share no copies, so this is only 8 × 9.

The Power of a Power Rule #

Copies of copies multiply the indices.

Raising a power to a power multiplies the indices, because copies come in groups

Squaring $2^3$2³ means two copies of the whole thing: $2^3\times 2^3$2³ × 2³.

Each copy unpacks into three 2s — six 2s in all. The indices multiplied: 3 × 2.

In general ${\left(a^m\right)}^n=a^{mn}$(aᵐ)ⁿ = aᵐⁿ — the one index law that multiplies the indices.

Side by side: multiplying powers adds the indices; a power of a power multiplies.

Zero and Negative Indices #

Count down past zero and you get reciprocals.

Counting the indices down past zero turns a power into its reciprocal

Step the index down by 1 and the value halves each time.

Carry on and $2^0$2⁰ has to be 1. Every base does.

One more step gives $\frac{1}{2}$1/2. A negative index means one over it.

Squares and Square Roots #

The root finds the side of the square back.

Squaring builds a square from a number, and the square root finds the side back

$5^2$5² is 5 × 5 = 25 — the tiles in a square with side 5.

$\sqrt{25}$√25 asks the backwards question: 25 tiles make a square with which side? 5.

The square numbers 1, 4, 9, 16, 25 are the ones whose roots come out whole.

Between squares the root is not whole: $\sqrt{20}$√20 lands between 4 and 5.

Cubes and Cube Roots #

Three equal copies, and the number behind them.

Cubing multiplies three copies of a number, and the cube root finds that number back

Cubed means three copies multiplied: 3 × 3 × 3 = 27 fills a 3 by 3 by 3 cube.

The cube root asks backwards: three copies of which number make 27? The answer is 3.

Two copies against three: 8 × 8 makes 64, and so does 4 × 4 × 4.

A cube of 64 unit cubes: every edge matches — the edge is the cube root, and $\sqrt[3]{64}$∛64 is 4.

Roots by Prime Factorization #

Split into primes, then share the copies out.

A root divides the prime copies out: halve the exponents for a square root, third them for a cube root

Split 144 into primes: 144 is 12 × 12, and each 12 breaks into 2 × 2 × 3.

A square root halves each exponent: half of the four 2s and half of the two 3s make 12.

A cube root keeps a third of each: two 2s and one 3 from 1728, so $\sqrt[3]{1728}$∛1728 is 12.

Powers in the Order of Operations #

Powers act before × and ÷, after brackets.

A power is worked out after any brackets and before any multiplying or dividing

The index binds to the 4 alone: $4^2$4² is 16, so $3\times 4^2$3 × 4² is 3 × 16, which is 48.

Brackets still outrank a power: ${\left(3\times 4\right)}^2$(3 × 4)² squares the 12, giving 144, not 48.

Adding waits its turn behind the power too: $5+2^3$5 + 2³ is 5 + 8, which is 13.

A root sits at the same rank: $\sqrt{36}$√36 becomes 6, then 6 × 2 is 12, and 20 − 12 is 8.

So the ladder gains a rung: brackets, then powers and roots, then × ÷, then + −.

Fractional Indices #

A half-power is a square root.

A fractional index is a root, because halving the index halves the multiplying

Adding the indices gives 1, so each half-power must be a square root.

So $9^{\frac{1}{2}}=3$9^½ = 3, the number that squares to 9.

A third-power is a cube root, and 8 comes from 2 × 2 × 2.

Standard Form #

One digit, a decimal part, a power of ten.

Standard form writes any number as one digit, a decimal part, and a power of ten

Move the point until one digit sits in front. That divided by 1000.

4.7 is 1000 times too small, so $\times 10^3$× 10³ restores it: $4700=4.7\times 10^3$4700 = 4.7 × 10³.

Sliding right multiplied by 1000, so ÷ 1000 puts it back: $0.0047=4.7\times 10^{-3}$0.0047 = 4.7 × 10⁻³.

Calculating in Standard Form #

Fronts multiply; powers of ten add.

Standard-form numbers multiply front with front while the powers of ten add

Multiplying: the fronts multiply, and the powers add — the index law from earlier.

Dividing mirrors it: fronts divide, powers subtract. 7 minus 3 leaves $10^4$10⁴.

Adding is the trap: match the powers first. $3\times 10^5$3 × 10⁵ is $30\times 10^4$30 × 10⁴, then 30 + 4.

If the front grows past 10, hand a ten to the power: $30\times 10^8$30 × 10⁸ is $3\times 10^9$3 × 10⁹.

Radicals #

A root left exact instead of rounded.

A radical is a root left exact instead of rounded, and it can often be simplified

The decimal for $\sqrt{2}$√2 never stops and never repeats, so writing $\sqrt{2}$√2 is the exact answer.

Split the number into a square factor and whatever is left.

The square part comes out of the root, and the rest stays in.

Check it: squaring $2\sqrt{3}$2√3 gives 4 × 3 = 12, so the split really did keep the value.

Rationalizing the Denominator #

No root left in the denominator.

Rationalizing multiplies numerator and denominator by the root to clear it

The value is fine, but this form is awkward to compare or add.

Multiply numerator and denominator by $\sqrt{2}$√2 — that is multiplying by 1.

In the denominator, $\sqrt{2}\times \sqrt{2}$√2 × √2 is exactly 2. The root is gone, which was the point.

Same move for $\frac{6}{\sqrt{3}}$6/√3: the denominator becomes 3, and 6 over 3 gives $2\sqrt{3}$2√3.

Adding and Multiplying Radicals #

Like terms, and one shared root.

A radical is a root left exact: radicals add as like terms, multiply under one root

3 of $\sqrt{2}$√2 and 5 more of $\sqrt{2}$√2 make 8 of it — like terms, with $\sqrt{2}$√2 in the letter seat.

Multiplying joins the roots: $\sqrt{2}\times \sqrt{8}=\sqrt{16}$√2 × √8 = √16 — and $\sqrt{16}$√16 is exactly 4.

Adding never joins different roots: $\sqrt{2}+\sqrt{3}$√2 + √3 simply stays as it is.

Two brackets expand as ever — four pieces: 2, $-\sqrt{3}$−√3, $2\sqrt{3}$2√3 and −3.

Collect: 2 − 3 makes −1, and $-\sqrt{3}+2\sqrt{3}$−√3 + 2√3 makes $\sqrt{3}$√3. The whole thing is $\sqrt{3}-1$√3 − 1.

Rational and Irrational Numbers #

Some numbers no fraction can reach.

A rational number is one whole number over another, and some numbers are not

A rational number is one whole number over another, like $\frac{3}{4}$3/4 or $\frac{1}{3}$1/3.

No fraction at all equals $\sqrt{2}$√2, so it is called irrational.

$\pi$π is irrational too, and $\frac{22}{7}$22/7 only comes close — it is not equal to $\pi$π.

A decimal that stops or repeats can always be turned back into a fraction.