Algebra 1 - (Chapter 7) Exponents and Exponential Functions - Free Printable

Here are the solutions to the problems on the worksheet. I have worked through each one step-by-step using the division properties of exponents.

Key Rule to Remember:
When dividing terms with the same base, you subtract the exponents: $\frac{x^a}{x^b} = x^{a-b}$.
Also, remember that any number raised to the power of 0 is 1 ($x^0 = 1$), and a negative exponent means the term belongs in the denominator ($x^{-n} = \frac{1}{x^n}$).

Step-by-Step Solutions

1. $\frac{3m^9}{m^7}$
* Coefficients: $3/1 = 3$
* Variables: $m^{9-7} = m^2$
* Result: $3m^2$

2. $\frac{-6c^4d^2}{2cd}$
* Coefficients: $-6 / 2 = -3$
* $c$ terms: $c^{4-1} = c^3$
* $d$ terms: $d^{2-1} = d^1 = d$
* Result: $-3c^3d$

3. $\frac{5y^2}{y^5}$
* Coefficients: $5/1 = 5$
* Variables: $y^{2-5} = y^{-3}$
* Move negative exponent to bottom: $\frac{5}{y^3}$

4. $\frac{28p^5q^3}{-7pq^3}$
* Coefficients: $28 / -7 = -4$
* $p$ terms: $p^{5-1} = p^4$
* $q$ terms: $q^{3-3} = q^0 = 1$ (they cancel out)
* Result: $-4p^4$

5. $\frac{a^2b^3c^4}{abc}$
* $a$ terms: $a^{2-1} = a$
* $b$ terms: $b^{3-1} = b^2$
* $c$ terms: $c^{4-1} = c^3$
* Result: $ab^2c^3$

6. $\frac{8x^2y^5z}{2xy^2z^3}$
* Coefficients: $8 / 2 = 4$
* $x$ terms: $x^{2-1} = x$
* $y$ terms: $y^{5-2} = y^3$
* $z$ terms: $z^{1-3} = z^{-2} = \frac{1}{z^2}$
* Result: $\frac{4xy^3}{z^2}$

7. $\left(\frac{a^2b^3}{ab}\right)^2$
* Simplify inside first: $\frac{a^2}{a} = a$, $\frac{b^3}{b} = b^2$. Inside becomes $ab^2$.
* Apply outer exponent: $(ab^2)^2 = a^2(b^2)^2 = a^2b^4$
* Result: $a^2b^4$

8. $\left(\frac{8x^6}{2x^2}\right)^3$
* Simplify inside first: $8/2 = 4$, $x^{6-2} = x^4$. Inside becomes $4x^4$.
* Apply outer exponent: $(4x^4)^3 = 4^3 \cdot (x^4)^3 = 64x^{12}$
* Result: $64x^{12}$

9. $\frac{(2x^3)^2}{4x^4}$
* Expand numerator: $(2x^3)^2 = 4x^6$
* Divide: $\frac{4x^6}{4x^4}$
* Coefficients: $4/4 = 1$
* Variables: $x^{6-4} = x^2$
* Result: $x^2$

10. $p^{-1} \cdot p^6 \cdot p^{-4}$
* When multiplying, add exponents: $-1 + 6 + (-4) = 1$
* Result: $p^1$ or just $p$

11. $x^3 \cdot x^{-5} \cdot x^{-1}$
* Add exponents: $3 + (-5) + (-1) = -3$
* Result: $x^{-3}$ or $\frac{1}{x^3}$

12. $12^0$
* Any non-zero number to the power of 0 is 1.
* Result: $1$

13. $\left(\frac{2}{3}\right)^{-2}$
* Negative exponent flips the fraction: $\left(\frac{3}{2}\right)^2$
* Square top and bottom: $\frac{3^2}{2^2} = \frac{9}{4}$
* Result: $\frac{9}{4}$

14. $\left(\frac{x^2}{y^3}\right)^{-1}$
* Flip the fraction: $\frac{y^3}{x^2}$
* Result: $\frac{y^3}{x^2}$

15. $12^{-2}$
* Make positive exponent by moving to denominator: $\frac{1}{12^2}$
* Calculate $12^2 = 144$
* Result: $\frac{1}{144}$

16. $\frac{-12u^5v^{-3}}{3u^{-2}v}$
* Coefficients: $-12 / 3 = -4$
* $u$ terms: $u^{5 - (-2)} = u^{5+2} = u^7$
* $v$ terms: $v^{-3 - 1} = v^{-4} = \frac{1}{v^4}$
* Result: $\frac{-4u^7}{v^4}$

17. $\frac{8x^2y^{-3}z^4}{4x^{-1}yz^2}$
* Coefficients: $8 / 4 = 2$
* $x$ terms: $x^{2 - (-1)} = x^3$
* $y$ terms: $y^{-3 - 1} = y^{-4} = \frac{1}{y^4}$
* $z$ terms: $z^{4 - 2} = z^2$
* Result: $\frac{2x^3z^2}{y^4}$

18. $\left(\frac{x^{-2}y^3}{z^{-1}}\right)^2$
* Apply square to everything: $x^{-4}y^6z^2$ (Note: $(z^{-1})^2 = z^{-2}$, which moves to top as $z^2$)
* Wait, let's do it carefully: Numerator becomes $x^{-4}y^6$. Denominator becomes $z^{-2}$.
* Expression: $\frac{x^{-4}y^6}{z^{-2}}$
* Move negative exponents: $x^{-4}$ goes down, $z^{-2}$ goes up.
* Result: $\frac{y^6z^2}{x^4}$

19. $\frac{6x^2y^{-3}z^5}{2x^{-1}y^2z}$
* Coefficients: $6 / 2 = 3$
* $x$ terms: $x^{2 - (-1)} = x^3$
* $y$ terms: $y^{-3 - 2} = y^{-5} = \frac{1}{y^5}$
* $z$ terms: $z^{5 - 1} = z^4$
* Result: $\frac{3x^3z^4}{y^5}$

20. $\frac{-12x^{-3}y^2z^{-1}}{4x^2y^{-1}z^3}$
* Coefficients: $-12 / 4 = -3$
* $x$ terms: $x^{-3 - 2} = x^{-5} = \frac{1}{x^5}$
* $y$ terms: $y^{2 - (-1)} = y^3$
* $z$ terms: $z^{-1 - 3} = z^{-4} = \frac{1}{z^4}$
* Result: $\frac{-3y^3}{x^5z^4}$

21. $\frac{e^2}{e^7}$
* Subtract exponents: $2 - 7 = -5$
* Result: $e^{-5}$ or $\frac{1}{e^5}$

22. $\frac{w^3x^4y}{wx^2y^5}$
* $w$ terms: $w^{3-1} = w^2$
* $x$ terms: $x^{4-2} = x^2$
* $y$ terms: $y^{1-5} = y^{-4} = \frac{1}{y^4}$
* Result: $\frac{w^2x^2}{y^4}$

23. $\frac{5a^{-2}b^3}{10a^4b^{-1}}$
* Coefficients: $5 / 10 = \frac{1}{2}$
* $a$ terms: $a^{-2 - 4} = a^{-6} = \frac{1}{a^6}$
* $b$ terms: $b^{3 - (-1)} = b^4$
* Result: $\frac{b^4}{2a^6}$

24. $\frac{2g^3h^{-2}j^4}{8g^{-1}h^3j}$
* Coefficients: $2 / 8 = \frac{1}{4}$
* $g$ terms: $g^{3 - (-1)} = g^4$
* $h$ terms: $h^{-2 - 3} = h^{-5} = \frac{1}{h^5}$
* $j$ terms: $j^{4 - 1} = j^3$
* Result: $\frac{g^4j^3}{4h^5}$

25. $\frac{(2a^2b)^3}{(4ab^2)^2}$
* Expand Top: $2^3(a^2)^3b^3 = 8a^6b^3$
* Expand Bottom: $4^2a^2(b^2)^2 = 16a^2b^4$
* Divide: $\frac{8a^6b^3}{16a^2b^4}$
* Coefficients: $8/16 = \frac{1}{2}$
* $a$ terms: $a^{6-2} = a^4$
* $b$ terms: $b^{3-4} = b^{-1} = \frac{1}{b}$
* Result: $\frac{a^4}{2b}$

26. BIOLOGY Problem
* Red blood cells: $22^5$
* White blood cells: $22^3$
* Question: How many times more red than white? This means divide red by white.
* Calculation: $\frac{22^5}{22^3}$
* Subtract exponents: $5 - 3 = 2$
* Result: $22^2$
* Calculate value: $22 \times 22 = 484$

──────────────────────────────────────

Final Answer:
1. $3m^2$
2. $-3c^3d$
3. $\frac{5}{y^3}$
4. $-4p^4$
5. $ab^2c^3$
6. $\frac{4xy^3}{z^2}$
7. $a^2b^4$
8. $64x^{12}$
9. $x^2$
10. $p$
11. $\frac{1}{x^3}$
12. $1$
13. $\frac{9}{4}$
14. $\frac{y^3}{x^2}$
15. $\frac{1}{144}$
16. $\frac{-4u^7}{v^4}$
17. $\frac{2x^3z^2}{y^4}$
18. $\frac{y^6z^2}{x^4}$
19. $\frac{3x^3z^4}{y^5}$
20. $\frac{-3y^3}{x^5z^4}$
21. $\frac{1}{e^5}$
22. $\frac{w^2x^2}{y^4}$
23. $\frac{b^4}{2a^6}$
24. $\frac{g^4j^3}{4h^5}$
25. $\frac{a^4}{2b}$
26. 484