Numerical Expressions with Fractions and Powers with Integer Exponents
Preparing for the Kyrgyzstan State Final Assessment (Grade 9)
What is a Numerical Expression?
A numerical expression is a combination of numbers and mathematical operations — addition, subtraction, multiplication, division, and exponentiation — that can be simplified (evaluated) to produce a single numerical value.
Examples of numerical expressions:
$3 + 5 \times 2$, $\dfrac{2^3 + 1}{5 - 2}$, $\left(\dfrac{3}{4}\right)^{-2} \cdot 6$
$3 + 5 \times 2$, $\dfrac{2^3 + 1}{5 - 2}$, $\left(\dfrac{3}{4}\right)^{-2} \cdot 6$
In this lesson, we focus on expressions that involve fractions and powers with integer exponents (positive, negative, and zero).
Integer Exponents — Key Definitions
Positive Exponent
$$a^n = \underbrace{a \cdot a \cdot a \cdots a}_{n \text{ times}}$$
Example: $3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81$
Zero Exponent
$$a^0 = 1 \quad (a \neq 0)$$
Example: $7^0 = 1$, $(-5)^0 = 1$
Negative Exponent
$$a^{-n} = \frac{1}{a^n} \quad (a \neq 0)$$
Example: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$
⚠️ Key Insight: A negative exponent does not make the result negative! It tells us to take the reciprocal (flip) of the base raised to the positive exponent.