8.4 Negative and Zero Exponent Lesson Plan for 9th - 10th Grade ... - Free Printable

Let’s work through this step by step.

This worksheet is about negative and zero exponents. The key rules you need to know are:

1. Any non-zero number raised to the power of 0 equals 1.
→ For example: \(5^0 = 1\), \((-3)^0 = 1\)

2. A negative exponent means “take the reciprocal and make the exponent positive.”
→ For example: \(x^{-n} = \frac{1}{x^n}\)
So, \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\)

3. When simplifying expressions with variables and negative exponents, move the term with the negative exponent to the other side of the fraction (numerator ↔ denominator) and change the sign of the exponent.

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Now let’s go part by part.

Part A: Evaluate each expression

We’ll compute each one using the rules above.

1. \(4^0 = 1\)
2. \((-2)^0 = 1\)
3. \(9^0 = 1\)
4. \((-7)^0 = 1\)
→ All these are just 1 because anything (non-zero) to the 0 power is 1.

5. \(5^{-1} = \frac{1}{5^1} = \frac{1}{5}\)
6. \(8^{-1} = \frac{1}{8}\)
7. \(10^{-1} = \frac{1}{10}\)
8. \((-4)^{-1} = \frac{1}{-4} = -\frac{1}{4}\)

9. \(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\)
10. \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\)
11. \(5^{-2} = \frac{1}{5^2} = \frac{1}{25}\)
12. \((-2)^{-3} = \frac{1}{(-2)^3} = \frac{1}{-8} = -\frac{1}{8}\)

So for Part A, answers are:
1. 1
2. 1
3. 1
4. 1
5. 1/5
6. 1/8
7. 1/10
8. -1/4
9. 1/9
10. 1/8
11. 1/25
12. -1/8

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Part B: Write each without negative or zero exponents

We rewrite each so all exponents are positive.

#### First column:

13. \(x^{-5} = \frac{1}{x^5}\)
14. \(y^{-3} = \frac{1}{y^3}\)
15. \(a^{-2}b = \frac{b}{a^2}\) ← only \(a^{-2}\) moves down
16. \(m^{-1}n^{-1} = \frac{1}{mn}\) ← both move down
17. \(\frac{x^{-2}}{y} = \frac{1}{x^2 y}\) ← numerator’s \(x^{-2}\) goes down
18. \(\frac{a}{b^{-3}} = a b^3\) ← denominator’s \(b^{-3}\) moves up as \(b^3\)
19. \(\frac{p^{-2}}{q^{-4}} = \frac{q^4}{p^2}\) ← p^{-2} goes down, q^{-4} comes up
20. \(\frac{r^{-1}s^{-2}}{t^{-3}} = \frac{t^3}{r s^2}\) ← r^{-1}, s^{-2} go down; t^{-3} comes up
21. \(\left( \frac{x}{y} \right)^{-2} = \left( \frac{y}{x} \right)^2 = \frac{y^2}{x^2}\) ← flip the fraction, then square
22. \(\left( \frac{a^{-1}}{b} \right)^{-3} = \left( \frac{b}{a^{-1}} \right)^3 = (a b)^3 = a^3 b^3\) ← first simplify inside: \(\frac{a^{-1}}{b} = \frac{1}{ab}\), then raise to -3 → \((ab)^3\)

Wait — let me double-check #22:

Original: \(\left( \frac{a^{-1}}{b} \right)^{-3}\)

Step 1: Simplify inside: \(a^{-1}/b = \frac{1}{a b}\)

Step 2: Raise to -3: \(\left( \frac{1}{ab} \right)^{-3} = (ab)^3 = a^3 b^3\) ✔

Alternatively, use rule: \(\left( \frac{A}{B} \right)^{-n} = \left( \frac{B}{A} \right)^n\)

So: \(\left( \frac{a^{-1}}{b} \right)^{-3} = \left( \frac{b}{a^{-1}} \right)^3 = (b \cdot a)^3 = a^3 b^3\) ✔

#### Second column:

23. \(z^{-4} = \frac{1}{z^4}\)
24. \(w^{-2} = \frac{1}{w^2}\)
25. \(c^{-1}d^{-1} = \frac{1}{cd}\)
26. \(e^{-3}f = \frac{f}{e^3}\)
27. \(\frac{u^{-4}}{v} = \frac{1}{u^4 v}\)
28. \(\frac{x}{y^{-2}} = x y^2\)
29. \(\frac{m^{-3}}{n^{-5}} = \frac{n^5}{m^3}\)
30. \(\frac{p^{-2}q^{-1}}{r^{-4}} = \frac{r^4}{p^2 q}\)
31. \(\left( \frac{m}{n} \right)^{-3} = \left( \frac{n}{m} \right)^3 = \frac{n^3}{m^3}\)
32. \(\left( \frac{x^{-2}}{y^{-1}} \right)^{-2} = \left( \frac{y^{-1}}{x^{-2}} \right)^2 = \left( \frac{x^2}{y} \right)^2 = \frac{x^4}{y^2}\)

Check #32 again:

Original: \(\left( \frac{x^{-2}}{y^{-1}} \right)^{-2}\)

Flip the fraction due to outer -2 exponent: \(\left( \frac{y^{-1}}{x^{-2}} \right)^2\)

Simplify inside: \(\frac{y^{-1}}{x^{-2}} = \frac{x^2}{y}\)

Then square it: \(\left( \frac{x^2}{y} \right)^2 = \frac{x^4}{y^2}\) ✔

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All done! Now compile final answers in order.

Final Answer:
Part A:
1. 1
2. 1
3. 1
4. 1
5. \(\frac{1}{5}\)
6. \(\frac{1}{8}\)
7. \(\frac{1}{10}\)
8. \(-\frac{1}{4}\)
9. \(\frac{1}{9}\)
10. \(\frac{1}{8}\)
11. \(\frac{1}{25}\)
12. \(-\frac{1}{8}\)

Part B:
13. \(\frac{1}{x^5}\)
14. \(\frac{1}{y^3}\)
15. \(\frac{b}{a^2}\)
16. \(\frac{1}{mn}\)
17. \(\frac{1}{x^2 y}\)
18. \(a b^3\)
19. \(\frac{q^4}{p^2}\)
20. \(\frac{t^3}{r s^2}\)
21. \(\frac{y^2}{x^2}\)
22. \(a^3 b^3\)
23. \(\frac{1}{z^4}\)
24. \(\frac{1}{w^2}\)
25. \(\frac{1}{cd}\)
26. \(\frac{f}{e^3}\)
27. \(\frac{1}{u^4 v}\)
28. \(x y^2\)
29. \(\frac{n^5}{m^3}\)
30. \(\frac{r^4}{p^2 q}\)
31. \(\frac{n^3}{m^3}\)
32. \(\frac{x^4}{y^2}\)