The image shows a table that demonstrates how to write expressions involving exponents in
expanded form, specifically for fractions where the numerator and denominator are powers of the same base.
Let’s go through each row and explain the pattern, then simplify the expressions using exponent rules.
---
Row 1:
$$
\frac{2^5}{2^3}
$$
Expanded Form:
$$
\frac{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2}{2 \cdot 2 \cdot 2}
$$
We can cancel out common factors (three 2s from numerator and denominator):
$$
= 2 \cdot 2 = 2^2
$$
✔ So:
$$
\frac{2^5}{2^3} = 2^{5-3} = 2^2
$$
---
Row 2:
$$
\frac{x^7}{x^4}
$$
Expanded Form:
$$
\frac{x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x}{x \cdot x \cdot x \cdot x}
$$
Cancel four $x$'s from numerator and denominator:
$$
= x \cdot x \cdot x = x^3
$$
✔ So:
$$
\frac{x^7}{x^4} = x^{7-4} = x^3
$$
---
Row 3:
$$
\frac{a^3}{a}
$$
Expanded Form:
$$
\frac{a \cdot a \cdot a}{a}
$$
Cancel one $a$:
$$
= a \cdot a = a^2
$$
✔ So:
$$
\frac{a^3}{a} = a^{3-1} = a^2
$$
---
✔ General Rule:
When dividing powers with the
same base, subtract the exponents:
$$
\frac{b^m}{b^n} = b^{m-n} \quad \text{(for } b \ne 0\text{)}
$$
---
Final Answers:
1. $\frac{2^5}{2^3} = 2^2$
2. $\frac{x^7}{x^4} = x^3$
3. $\frac{a^3}{a} = a^2$
These simplified forms follow directly from the
Quotient Rule of Exponents.
Let me know if you'd like to see how this applies to more complex expressions!
Parent Tip: Review the logic above to help your child master the concept of quotient of powers worksheet.
