Powers and Roots

Index Notation #

The little number counts the copies.

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

Read the full lesson →

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

Read the full lesson →

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

Read the full lesson →

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

Read the full lesson →

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

Read the full lesson →

$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

Read the full lesson →

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

Read the full lesson →

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

Read the full lesson →

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

Read the full lesson →

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

Read the full lesson →

Start with one digit before the point: 4.7. Multiplying by 1000 moves each digit three places left, back to 4700.

Three places is × 1000, and $1000=10^3$1000 = 10³, so $4700=4.7\times 10^3$4700 = 4.7 × 10³.

A small number moves the other way: 4.7 ÷ 1000 = 0.0047, so $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

Read the full lesson →

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

Read the full lesson →

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

Read the full lesson →

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

Read the full lesson →

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

Read the full lesson →

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.