Exponent Laws

In Folding Paper to the Moon, the layer count doubled past every estimate in the room, and keeping track meant multiplying powers of 2 at speed. The laws below are not new facts to memorise — each one is just counting factors, and if you can rebuild a law in ten seconds you never need to trust your memory of it.

Laws you can rebuild

Law	In symbols	Rebuild it by
Product	$x^a\cdot x^b=x^{a+b}$	counting factors: $a$ of them, then $b$ more
Quotient	$x^a÷x^b=x^{a-b}$	cancelling $b$ factors from the top
Power of a power	${\left(x^a\right)}^b=x^{ab}$	$a$ factors, written down $b$ times

The third law is quietly the most useful in this course: it lets one function wear different bases. Since $9=3^2$,

$$9^x={\left(3^2\right)}^x=3^{2x}$$

— the same function in different clothes, a costume change you will use when comparing exponential models.

Zero and below

What should $2^0$ mean? Do not decree it — descend to it. Each step down the list divides by 2:

$$2^3=8,2^2=4,2^1=2,2^0=1,2^{-1}=\frac{1}{2},2^{-2}=\frac{1}{4}$$

The pattern forces $2^0=1$, and keeps going: a negative exponent means reciprocal, never “negative number”. So $2^{-3}=\frac{1}{8}$, a small positive number. This descend-the-pattern move shows up regularly in Number Strings — by the third time you run it, the laws feel inevitable rather than imposed.

Exponent Laws Practice mixes numeric and algebraic work, and Rational Exponents takes the next step down this same road: what must $4^{1/2}$ mean?

Curriculum connection

B1.3

simplify algebraic expressions containing integer and rational exponents [e.g., $(x^3)÷(x^{1/2})$, $(x^6{y}^3{)}^{1/3}$], and evaluate numeric expressions containing integer and rational exponents and rational bases [e.g., $2^{-3}$, $(-6{)}^3$, $4^{1/2}$, $1.01^{120}$]

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B2.4

determine, through investigation using technology, that the equation of a given exponential function can be expressed using different bases [e.g., $f(x)=9^x$ can be expressed as $f(x)=3^{2x}$], and explain the connections between the equivalent forms in a variety of ways (e.g., comparing graphs; using transformations; using the exponent laws)

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