Exponents & Exponential Functions

A1.G.1Exponent properties01/05

Exponent properties — review.

Step01of 05

An exponent says "multiply the base by itself this many times": $2^{4} = 2 \cdot 2 \cdot 2 \cdot 2 = 16$ . Every exponent rule below comes from that one definition by counting copies.

Product of powers	→ $x^{a} \cdot x^{b} = x^{a + b}$
Quotient of powers	→ $\frac{x ^{a}}{x ^{b}} = x^{a - b}$
Power of a power	→ $(x^{a})^{b} = x^{a \cdot b}$
Power of a product	→ $(x y)^{a} = x^{a} y^{a}$

A1.G.1 — Exponent properties — review.

Step 01 of 05

An exponent says "multiply the base by itself this many times": $2^{4} = 2 \cdot 2 \cdot 2 \cdot 2 = 16$ . Every exponent rule below comes from that one definition by counting copies.

Product of powers	→ $x^{a} \cdot x^{b} = x^{a + b}$
Quotient of powers	→ $\frac{x ^{a}}{x ^{b}} = x^{a - b}$
Power of a power	→ $(x^{a})^{b} = x^{a \cdot b}$
Power of a product	→ $(x y)^{a} = x^{a} y^{a}$

Step 02 of 05

Two definitions you must memorize — they aren't proofs, they're the conventions that make the other rules consistent.

x^{0} = 1 (x \neq = 0) x^{- n} = \frac{1}{x ^{n}}

Why $x^{0} = 1$ ? Because $x^{a} / x^{a} = x^{a - a} = x^{0}$ and that quotient is clearly $1$ .

Step 03 of 05

Worked example — combine all four rules. Simplify:

\frac{( 2 x ^{3} ) ^{2} \cdot x ^{4}}{x ^{5}}

Step 1 — power of a power on the numerator's first piece.

(2 x^{3})^{2} = 2^{2} \cdot x^{3 \cdot 2} = 4 x^{6}

Step 04 of 05

Step 2 — combine the numerator using product of powers.

4 x^{6} \cdot x^{4} = 4 x^{6 + 4} = 4 x^{10}

Step 3 — divide using quotient of powers.

\frac{4 x ^{10}}{x ^{5}} = 4 x^{10 - 5} = 4 x^{5}

Step 05 of 05

Common traps.

$(x + y)^{2}$	→NOT $x^{2} + y^{2}$ — power doesn't distribute over addition
$- 3^{2}$ vs. $(- 3)^{2}$	→first is $- 9$ , second is $9$ — exponent binds tighter than negative
$x^{a} + x^{b}$	→can NOT combine — only multiplication adds exponents

Key insight

Exponent rules add or subtract exponents when you MULTIPLY or DIVIDE powers with the same base. They never simplify a sum or difference of powers.

A1.G.2 — Rational exponents.

Step 01 of 05

Fractional exponents extend the rules you already know to roots. The key definition:

x^{1/ n} = n x

$x^{1/2}$ is the square root, $x^{1/3}$ is the cube root, and so on. Why? Because we want $x^{1/2} \cdot x^{1/2} = x^{1} = x$ — the only number that squares to x is $x$ .

Step 02 of 05

Generalize to any rational exponent.

x^{m / n} = n x^{m} = (n x)^{m}

The numerator is the power; the denominator is the root index. You can apply them in either order — root first is usually easier with small numbers.

8^{2/3} = (38)^{2} = 2^{2} = 4

Step 03 of 05

Worked example — convert and simplify.

2 7^{4/3}

Take the cube root first (denominator), then raise to the 4th (numerator):

2 7^{4/3} = (327)^{4} = 3^{4} = 81

Going the other direction works too but blows up the arithmetic: $2 7^{4} = 531, 441$ , then cube root. Pick root-first when you can.

Step 04 of 05

Negative rational exponents combine two rules.

x^{- m / n} = \frac{1}{x ^{m / n}}

1 6^{- 3/4} = \frac{1}{1 6 ^{3/4}} = \frac{1}{( 4 16 ) ^{3}} = \frac{1}{2 ^{3}} = \frac{1}{8}

The negative flips, the rational exponent does the root + power.

Step 05 of 05

All the exponent rules from skill 1 still apply. The exponents just happen to be fractions now.

x^{1/2} \cdot x^{3/2} = x^{1/2 + 3/2} = x^{2}

(x^{2/3})^{6} = x^{2/3 \cdot 6} = x^{4}

Key insight

Rational exponents = roots in disguise.

x^{m / n}

means "take the n-th root, raise to the m-th power." Every other exponent rule still works.

A1.G.3 — Exponential growth and decay models.

The general form

y = a \cdot b^{t}

Step 01 of 05

An exponential function multiplies by the same factor in every step. The general model:

a	→initial value (the output when $t = 0$ )
b	→growth/decay factor per unit of time
$t$	→time elapsed
$b > 1$	→growth
$0 < b < 1$	→decay

Step 02 of 05

Growth example. A bacteria colony starts at 200 and triples every hour. Model the population after $t$ hours.

y = 200 \cdot 3^{t}

Initial value $a = 200$ , factor $b = 3$ (triples, so multiply by 3 each step). After 4 hours:

y = 200 \cdot 3^{4} = 200 \cdot 81 = 16, 200

Step 03 of 05

Decay example. A car worth $24,000 loses 15% of its value every year. Model the value after $t$ years.

y = 24000 \cdot (0.85)^{t}

The factor $b = 0.85 = 1 - 0.15$ — keeping 85% IS losing 15%. After 5 years:

y = 24000 \cdot (0.85)^{5} \approx 24000 \cdot 0.4437 \approx $10, 649

Step 04 of 05

Translating "increases by r%" into the formula.

Increases by r%	→ $b = 1 + r$ (with r as a decimal)
Decreases by r%	→ $b = 1 - r$
Doubles	→ $b = 2$
Halves	→ $b = \frac{1}{2}$
Triples	→ $b = 3$

"Grows by 8%" means $b = 1.08$ . "Loses 30%" means $b = 0.70$ . Get the FACTOR right and the rest is plug-and-chug.

Step 05 of 05

Reading a model. Given $P = 50 (1.06)^{t}$ for population after $t$ years:

Initial value	→ $50$ (the $a$ )
Annual growth rate	→ $6%$ (since $1.06 = 1 + 0.06$ )
Doubling time	→solve $100 = 50 (1.06)^{t}$ for $t$

Key insight

Two quantities tell you everything: the initial value a and the per-step factor b. From those, every future value is one substitution away.

A1.G.4 — Compound interest.

The compound interest formula

A = P (1 + \frac{r}{n})^{n t}

Step 01 of 05

Compound interest is exponential growth applied to money.

$A$	→final amount
$P$	→principal (starting amount)
r	→annual interest rate (as a decimal)
$n$	→compoundings per year (1 = annual, 4 = quarterly, 12 = monthly)
$t$	→years

The reason it's not just $P (1 + r)^{t}$ : when interest compounds MORE often than yearly, each compounding adds a smaller fraction ( $r / n$ ) but does so more times ( $n t$ ).

Step 02 of 05

Worked example. $1,000 at 6% annual interest, compounded quarterly, for 5 years.

P = 1000, r = 0.06, n = 4, t = 5

Plug in:

A = 1000 (1 + \frac{0.06}{4})^{4 \cdot 5} = 1000 (1.015)^{20}

Step 03 of 05

Evaluate the exponent and finish.

(1.015)^{20} \approx 1.3469

A \approx 1000 \cdot 1.3469 \approx $1, 346.86

Same scenario compounded ANNUALLY ( $n = 1$ ) gives $1000 (1.06)^{5} \approx $1, 338.23$ . More frequent compounding earns a bit more — but the effect tapers fast.

Step 04 of 05

The effect of $n$ . Same $1,000 at 6% for 5 years:

Annually (n=1)	→$1,338.23
Quarterly (n=4)	→$1,346.86
Monthly (n=12)	→$1,348.85
Daily (n=365)	→$1,349.83

Diminishing returns. Going from annual → monthly matters; going from monthly → daily barely moves the needle.

Step 05 of 05

Reading a problem. The hard part is mapping words to symbols.

"Quarterly"	→ $n = 4$
"Monthly"	→ $n = 12$
"Semiannually"	→ $n = 2$
"5% annual rate"	→ $r = 0.05$
"$2000 invested"	→ $P = 2000$

Key insight

Compound interest is the exponential growth model with a specific shape: base

(1 + r / n)

, exponent

n t

. Identify each piece from the problem, then it's one calculator press.

A1.G.5 — Compare linear and exponential growth.

Step 01 of 05

Linear growth ADDS the same amount each step. Exponential growth MULTIPLIES by the same factor. Both feel similar early on, but exponential pulls away dramatically.

Linear	→ $y = m x + b$ — constant rate of change
Exponential	→ $y = a \cdot b^{x}$ — constant ratio between consecutive outputs

Step 02 of 05

Spot the type from a table. Look at consecutive outputs.

Constant difference	→linear, slope = the difference
Constant ratio	→exponential, factor = the ratio
Neither	→not linear, not exponential

Example table:

x : 0, 1, 2, 3 y : 3, 6, 12, 24

Differences: $3, 6, 12$ — not constant. Ratios: $6/3 = 2, 12/6 = 2, 24/12 = 2$ — constant. EXPONENTIAL, $y = 3 \cdot 2^{x}$ .

Step 03 of 05

The dramatic comparison. Start both at 1. Linear adds 5 per step; exponential doubles each step.

Step 0	→1 vs. 1
Step 5	→26 vs. 32 — exponential just passed
Step 10	→51 vs. 1,024
Step 20	→101 vs. 1,048,576

At step 20, exponential is over 10,000× larger. This is why "compounding wins" in finance and population models.

Step 04 of 05

The reverse — when does linear win? Early on, linear can stay ahead if its rate is large vs. a slow exponential. Eventually exponential ALWAYS overtakes — there's no rate of linear growth that beats exponential forever.

In Algebra I problems, the exponential function eventually dominates any linear function for large enough x. That's the general rule.

Step 05 of 05

Picking the right model from a description.

"Earns $50 each week"	→linear, m = 50
"Doubles every 3 days"	→exponential
"Population grows by 200/yr"	→linear
"Population grows by 4% / yr"	→exponential, b = 1.04
"Half-life of 10 days"	→exponential decay, b = 0.5 per 10 days

Key insight

"Per-step amount" → linear. "Per-step percentage or factor" → exponential. The wording almost always tells you which family the problem belongs to.

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