Real Numbers & Exponents

8.A.1Rational vs irrational01/04

Rational vs. irrational numbers.

Step01of 04

Real numbers split into two camps based on whether they can be written as a fraction.

Rational	→can be written as $p / q$ with integer $p, q$ and $q \neq = 0$
Irrational	→cannot — decimal expansion never terminates AND never repeats

8.A.1 — Rational vs. irrational numbers.

Step 01 of 04

Real numbers split into two camps based on whether they can be written as a fraction.

Rational	→can be written as $p / q$ with integer $p, q$ and $q \neq = 0$
Irrational	→cannot — decimal expansion never terminates AND never repeats

Step 02 of 04

Examples — rational.

Integers (5, -7, 0)	→e.g. $5 = 5/1$
Fractions (3/4, -2/9)	→already in p/q form
Terminating decimals (0.25)	→ $= 1/4$
Repeating decimals (0.3̄)	→ $= 1/3$

Examples — irrational.

$2 \approx 1.414213 \dots$	→square root of any non-perfect-square
$π \approx 3.14159 \dots$	→famous transcendental constant
$e \approx 2.71828 \dots$	→natural log base
$35, 7, 50$	→most roots are irrational

Step 03 of 04

How to tell them apart from a decimal.

Terminates	→rational
Repeats with a pattern	→rational
Never terminates AND never repeats	→irrational

Square root test. A square root of an integer is rational ONLY if that integer is a perfect square (1, 4, 9, 16, 25, …). Otherwise irrational.

Step 04 of 04

Operations.

Rational + rational	→rational
Rational + irrational	→irrational (e.g. $2 + 3$ )
Irrational + irrational	→could be either: $2 + (- 2) = 0$ (rational), but $2 + 3$ stays irrational
Rational × irrational (nonzero rational)	→irrational

Key insight

Rational = "ratio" of integers, equivalent to terminating-or-repeating decimals. Irrational = neither terminates nor repeats. Square roots of non-perfect-squares are irrational;

π

and

e

are the famous constants.

8.A.2 — Approximate irrational numbers on a number line.

Step 01 of 04

Irrationals don't have a clean decimal form, but you can pin them down between consecutive whole numbers (and finer) by squaring nearby values.

Example: $50$ . $7^{2} = 49$ , $8^{2} = 64$ , so $7 < 50 < 8$ . Closer to 7 (since 50 is closer to 49).

Step 02 of 04

Refine to one decimal place. Try squares of 7.0, 7.1, 7.2, …

$7. 0^{2} = 49$	→too low
$7. 1^{2} = 50.41$	→too HIGH (just slightly)
So	→ $7.0 < 50 < 7.1$

$50$ is between 7.0 and 7.1 — closer to 7.1 since $7. 1^{2} = 50.41$ is just barely above 50.

Step 03 of 04

For $π$ : memorize a few digits — $3.14159 \dots$ . For most class problems, $π \approx 3.14$ works.

For arbitrary irrationals, your calculator's decimal display gives an approximation. The exact value is the irrational expression itself ( $50 = 52$ ); the decimal is the approximation.

Step 04 of 04

Place on a number line. Once you know $50 \approx 7.07$ , tick marks between 7 and 8 let you mark it visually — slightly past 7, about 1/14 of the way to 8.

Compare two irrationals. Approximate each, then compare. Or, square them: if $a, b > 0$ , then $a < b \Leftrightarrow a^{2} < b^{2}$ .

Key insight

Approximate by squeezing: find perfect squares above and below, refine with decimals if needed. Calculator gives the decimal approximation; the exact value remains the irrational expression.

8.A.3 — Properties of integer exponents.

Step 01 of 04

The four core exponent properties — same as in algebra, just with integer exponents.

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

Step 02 of 04

Two definitions to memorize.

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

These extend the rules to zero and negative exponents in a way that keeps the four properties consistent.

Step 03 of 04

Worked examples.

$5^{3} \cdot 5^{2}$	→ $5^{5} = 3125$
$2^{7} / 2^{4}$	→ $2^{3} = 8$
$(3^{2})^{4}$	→ $3^{8} = 6561$
$4^{- 2}$	→ $1/16$
$(- 2)^{3}$	→ $- 8$
$- 2^{3}$	→ $- 8$ (same — exponent before negation only matters with parens)

Step 04 of 04

Common pitfalls.

$x^{a} + x^{b}$	→cannot combine — exponent rules require multiplication or division of powers
$(x + y)^{2}$	→NOT $x^{2} + y^{2}$ — must FOIL
$x^{- 1}$	→ $= 1/ x$ (NOT a negative number — just a reciprocal)

Key insight

Same exponent rules as in algebra.

x^{0} = 1

and

x^{- n} = 1/ x^{n}

extend them to all integers. Negative exponents create reciprocals; they don't make values negative.

8.A.4 — Square roots and cube roots.

Step 01 of 04

Square root. $a$ is the non-negative number that, when squared, gives $a$ . $a^{2} = ∣ a ∣$ .

Perfect squares to memorize.

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, \dots

Step 02 of 04

Cube root. $3 a$ is the number whose cube is $a$ . Unlike square roots, cube roots can be negative.

38 = 2, 3 - 27 = - 3

Perfect cubes to memorize.

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, \dots

Step 03 of 04

Solving equations.

$x^{2} = 49$	→ $x = \pm 7$ (BOTH solutions)
$x^{3} = 64$	→ $x = 4$ (one solution; cubing is one-to-one)
$x^{2} = - 16$	→no real solution (squares are never negative)
$x^{3} = - 27$	→ $x = - 3$

Step 04 of 04

Estimating non-perfect roots. Find perfect roots above and below, estimate based on which is closer.

$30$ : $25 = 5$ , $36 = 6$ , so $5 < 30 < 6$ — closer to 5 since 30 is much closer to 25.

Key insight

Square roots:

x^{2} = c

has TWO solutions

\pm c

. Cube roots:

x^{3} = c

has ONE solution

3 c

(cube roots can be negative). Memorize perfect squares to 15² and perfect cubes to 10³.

8.A.5 — Scientific notation — convert.

Step 01 of 04

Scientific notation writes any positive number as $a \times 1 0^{n}$ where $1 \leq a < 10$ and $n$ is an integer.

Useful for very large or very small numbers — the speed of light, the mass of an electron, the U.S. national debt, atomic radii.

Step 02 of 04

Standard → scientific. Move the decimal so only one nonzero digit is to its left. Count the moves; that's the exponent.

$5, 300, 000$	→ $5.3 \times 1 0^{6}$ (decimal moved 6 places left)
$0.000079$	→ $7.9 \times 1 0^{- 5}$ (decimal moved 5 places right)
$72$	→ $7.2 \times 1 0^{1}$
$0.5$	→ $5 \times 1 0^{- 1}$

Decimal moves LEFT → exponent positive. Decimal moves RIGHT → exponent negative.

Step 03 of 04

Scientific → standard. Move the decimal as many places as the exponent (right for positive, left for negative).

$6.02 \times 1 0^{23}$	→ $602, 000, 000, 000, 000, 000, 000, 000$
$3.14 \times 1 0^{- 4}$	→ $0.000314$

Pad with zeros as needed when you run out of digits.

Step 04 of 04

Compare two numbers in scientific notation.

Different exponents	→larger exponent wins (regardless of mantissa)
Same exponent	→compare the mantissas (the a part)

Example: $3.5 \times 1 0^{7}$ vs. $9.9 \times 1 0^{6}$ . The first is bigger because $1 0^{7} > 1 0^{6}$ , even though 9.9 > 3.5.

Key insight

Scientific notation =

a \times 1 0^{n}

, with

1 \leq a < 10

. Decimal moves LEFT for big numbers (positive exponent), RIGHT for small ones (negative). Compare exponents first; mantissas only as tiebreaker.

8.A.6 — Operations with scientific notation.

Step 01 of 04

Multiplication and division. Multiply (or divide) the mantissas. Add (or subtract) the exponents. Adjust if the result's mantissa falls outside $[1, 10)$ .

(3 \times 1 0^{4}) (2 \times 1 0^{5}) = 6 \times 1 0^{9}

\frac{8 \times 1 0 ^{7}}{2 \times 1 0 ^{3}} = 4 \times 1 0^{4}

Step 02 of 04

Adjust if needed.If the resulting mantissa is &geq;10 or <1, shift the decimal and adjust the exponent.

(5 \times 1 0^{3}) (4 \times 1 0^{2}) = 20 \times 1 0^{5} = 2 \times 1 0^{6}

(2 \times 1 0^{4}) (3 \times 1 0^{- 7}) = 6 \times 1 0^{- 3}

Step 03 of 04

Addition and subtraction are trickier — exponents must match before you can combine.

$3 \times 1 0^{4} + 5 \times 1 0^{4} = 8 \times 1 0^{4}$ . Same exponent, just add mantissas.

$3 \times 1 0^{4} + 5 \times 1 0^{3}$ : rewrite $5 \times 1 0^{3} = 0.5 \times 1 0^{4}$ , then $3 + 0.5 = 3.5 \times 1 0^{4}$ .

Step 04 of 04

Real-world example. Earth's distance to the sun: $1.5 \times 1 0^{8}$ km. Light speed: $3 \times 1 0^{5}$ km/sec. Time for sunlight to reach Earth:

\frac{1.5 \times 1 0 ^{8}}{3 \times 1 0 ^{5}} = 0.5 \times 1 0^{3} = 5 \times 1 0^{2} = 500 seconds

About 8 minutes 20 seconds — the famous "8 light-minutes" to Earth.

Key insight

Multiply / divide: combine mantissas, add or subtract exponents, normalize if mantissa drifts outside

[1, 10)

. Add / subtract: align exponents first, then combine. Practice with real-world examples to build intuition.

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