Exponent rules — multiplication, division, power of power, zero and negative exponents

Question

Simplify: $\dfrac{2^5 \times 3^4 \times 2^{-3}}{3^2 \times 2^4}$ . Also explain why $a^0 = 1$ for any non-zero $a$ .

(CBSE 7 & 8 — exponents chapter)

Solution — Step by Step

Rule	Formula	Example
Product (same base)	$a^m \times a^n = a^{m+n}$	$2^3 \times 2^4 = 2^7$
Quotient (same base)	$a^m \div a^n = a^{m-n}$	$5^6 \div 5^2 = 5^4$
Power of power	$(a^m)^n = a^{mn}$	$(3^2)^4 = 3^8$
Zero exponent	$a^0 = 1$ (if $a \neq 0$ )	$7^0 = 1$
Negative exponent	$a^{-n} = 1/a^n$	$2^{-3} = 1/8$
Power of product	$(ab)^n = a^n b^n$	$(2 \times 3)^4 = 2^4 \times 3^4$

\frac{2^5 \times 3^4 \times 2^{-3}}{3^2 \times 2^4}

Group the 2s and 3s:

= \frac{2^{5+(-3)} \times 3^4}{2^4 \times 3^2} = \frac{2^2 \times 3^4}{2^4 \times 3^2}

= 2^{2-4} \times 3^{4-2} = 2^{-2} \times 3^2 = \frac{1}{4} \times 9 = \mathbf{\frac{9}{4}}

From the quotient rule: $a^n \div a^n = a^{n-n} = a^0$ . But $a^n \div a^n = 1$ (any number divided by itself). So $a^0 = 1$ .

This only works when $a \neq 0$ because $0^0$ is undefined.

Why This Works

Exponents are repeated multiplication. The rules are shortcuts for combining these multiplications. For example, $a^3 \times a^2 = (a \times a \times a) \times (a \times a) = a^5$ — we just add the counts.

graph TD
    A["Exponent Problem"] --> B{"Same base?"}
    B -->|"Yes, multiplying"| C["ADD exponents<br/>a^m × a^n = a^(m+n)"]
    B -->|"Yes, dividing"| D["SUBTRACT exponents<br/>a^m ÷ a^n = a^(m-n)"]
    B -->|"Power of power"| E["MULTIPLY exponents<br/>(a^m)^n = a^(mn)"]
    B -->|"Different bases"| F["Cannot combine directly<br/>Compute each separately"]
    A --> G{"Negative exponent?"}
    G -->|"Yes"| H["Flip: a^(-n) = 1/a^n"]
    A --> I{"Zero exponent?"}
    I -->|"Yes"| J["Result = 1<br/>(if base ≠ 0)"]

Alternative Method — Convert Everything to Positive Exponents First

Move all negative exponents to the other side of the fraction:

\frac{2^5 \times 3^4 \times 2^{-3}}{3^2 \times 2^4} = \frac{2^5 \times 3^4}{3^2 \times 2^4 \times 2^3} = \frac{2^5 \times 3^4}{2^7 \times 3^2} = \frac{3^2}{2^2} = \frac{9}{4}

For speed: always combine same bases first, then simplify. Don’t compute large powers — work with the exponents as numbers. $2^{15}$ is hard to calculate, but $2^{15-13} = 2^2 = 4$ is trivial.

Common Mistake

Students write $2^3 \times 3^2 = 6^5$ — you CANNOT add exponents when the bases are different. The product rule works only for the same base. $2^3 \times 3^2 = 8 \times 9 = 72$ , not $6^5 = 7776$ . This is the single most common exponent error in CBSE Class 7-8.

Question

Solution — Step by Step

Why This Works

Alternative Method — Convert Everything to Positive Exponents First

Common Mistake

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