HiSET Math Study Guide: Numbers & Operations

Numbers and operations questions will assess your ability to:

• Identify and classify rational and irrational numbers
• Solve problems using radicals and exponents
• Perform operations with numbers in scientific notation
• Solve multistep real-world math problems

Practice Quiz

Rational Numbers

Rational numbers can be written as a fraction $\frac{a}{b}$, where $a$ and $b$ are integers and $b ≠ 0$. In decimal form they must terminate $(.75)$ or repeat $(0.333…)$.

• Examples: $\frac{1}{2}, \;$ $-3, \;$ $4.75, \;$ $0.666… (\frac{2}{3})$

Irrational Numbers

Irrational numbers cannot be written as simple fractions. In decimal form, they never terminate or repeat.

• Examples: $\sqrt{2},\;$ $π \text{ (pi)},\;$ $1.6180339887…$

Understanding Exponents

An exponent tells you how many times to multiply a number by itself. It consists of two parts:

• The base, which is the number being multiplied.
• The exponent (or power), which tells how many times the base is used as a factor.

Let’s take a look at an example…

• Example: $3^4$:

In this example, 3 is the base and 4 is the exponent, meaning multiply 3 by itself 4 times:

$3^4=3×3×3×3=81$

Exponents provide a shortcut to writing repeated multiplication. Instead of writing $5×5×5×5×5$, we simply write $5^5$. This is useful in algebra, science, and real-world applications like computing large values in physics and finance.

Exponents Rules

Product Rule: $a^m \cdot a^n = a^{m+n}$

Quotient Rule: $\dfrac{a^m}{a^n} = a^{m-n}$

Power Rule: $(a^m)^n = a^{m \cdot n}$

Zero Exponent: $a^0 = 1 \, (\text{if } a \neq 0$)

Negative Exponent: $a^{-n} = \dfrac{1}{a^n}$

Understanding Radicals in a Simple Way

A radical (√) is a way to undo an exponent. It helps us find what number was multiplied by itself to get a given value.

Square Roots

The square root of a number asks: What number times itself equals this number?

• Example: $\sqrt{9} =3$ because $3×3=9$
• Example: $\sqrt{16} ​=4$ because $4×4=16$

Cube Roots and Higher Roots

The cube root asks: What number multiplied by itself three times equals this number?

• Example: $\sqrt[3]{8} = 2$ because $2×2×2=8$
• Example: $3\sqrt[3]{27} = 27$ because $3×3×3=27$

You can also have fourth roots, fifth roots, and more:

• Example: $\sqrt[4]{16} = 2$ because $2×2×2×2=16$

Hint
If a number under a square root is a perfect square (like 4, 9, 16, 25), it simplifies to a whole number. If not, it stays as a radical or a decimal.

Scientific Notation

Scientific notation is a way to write very large or very small numbers using powers of 10.

Basic Form

$a×10^n$

• $a$ is a number between 1 and 10 (but not 10).
• $n$ is an exponent that tells us how many places to move the decimal.

Converting to Scientific Notation

1. Move the decimal so that there is one nonzero digit to the left of it.
2. Count how many places the decimal moved—this is the exponent $n$.

If the original number is large, $n$ is positive.

If the original number is small, $n$ is negative.

• Example: $45{,}000=4.5×10^4$ (moved 4 places left)
• Example: $0.00032=3.2×10^{−4}$ (moved 4 places right)

Operations with Scientific Notation

Multiplication

$(a×10^m)×(b×10^n)= (a×b)×10^{m+n}$

Multiply the numbers ($a$ values). Add the exponents.

• Example: $(3×10^2)×(2×10^3)=6×10^5$

Division

$\dfrac{a \times 10^m}{b \times 10^n} = \left( \dfrac{a}{b} \right) \times 10^{m-n}$

Divide the numbers ($a$ values). Subtract the exponents.

• Example: $\dfrac{6 \times 10^5}{2 \times 10^2} = 3 \times 10^3$

Addition and Subtraction

Exponents must be the same before adding or subtracting.

If they are different, adjust one number so both have the same exponent.

• Example: $(3.2×10^4)+(4.5×10^3)$

Rewrite $4.5×10^3$ as $0.45×10^4$, then:

$(3.2+0.45)×10^4 = 3.65 \times 10^4$

Hint
For multiplication and division, work with the numbers and exponents separately. For addition and subtraction, make sure the exponents match first.

Numbers & Operations Review Quiz