Prealgebra for a 15-year-old: Chapter Overview (ISBN 978-1-934124-21-5) and a Clear, Step-by-Step Sample Lesson on Exponents

Clear, student-friendly prealgebra guide for a 15-year-old mapped to the listed chapters and standards. Includes study plan, topic choices, and a step-by-step mini-lesson on exponents with worked examples and practice problems.

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Quick note

I see the full chapter list for the Prealgebra book (ISBN 978-1-934124-21-5). Tell me which chapter or specific topic you want to learn now. Below is a short plan for using the book and a sample, step-by-step lesson on Chapter 2 (Exponents) so you can see how I teach topics to a 15-year-old student.

How we’ll use the book (simple plan)

  1. Pick one focused topic or chapter (you can pick a problem, too).
  2. Start with the definitions and rules—make sure they make sense logically.
  3. See worked examples step-by-step (I’ll show each small move and why it’s allowed).
  4. Try short practice problems (I’ll give hints if needed).
  5. Do a few mixed problems to connect this topic to others (e.g., exponents + fractions).
  6. Review common mistakes and a quick checklist for solving similar problems later.

Pick a topic now

Reply with one of the chapter names or a specific question (for example: "Chapter 4 Fractions: how to add mixed numbers" or "Chapter 12: proof of the Pythagorean theorem" or a specific exercise).

Sample mini-lesson: Chapter 2 — Exponents (step-by-step)

1) What is an exponent?

An exponent tells how many times to multiply a number by itself. If a is a number and n is a positive integer, then an means a × a × ... × a (n factors).

2) Notation and a few examples

  • 23 = 2 × 2 × 2 = 8
  • 51 = 5 (anything to the 1st power is itself)
  • 70 = 1 (more on zero exponent below)

3) Exponent laws (the rules) — each with a short reason

  1. Product rule: am · an = am+n
    Reason: multiplying m copies of a with n more copies gives m+n copies.
  2. Quotient rule: am ÷ an = am−n (for a ≠ 0)
    Reason: cancel common copies of a in numerator and denominator.
  3. Power of a power: (am)n = am·n
    Reason: you multiply groups of m copies n times, total m·n copies.
  4. Power of a product: (ab)n = an bn
    Reason: distribute the repeated multiplication to each factor.
  5. Zero exponent: a0 = 1 for a ≠ 0
    Reason: use quotient rule: an ÷ an = an−n = a0, but left side is 1, so a0 = 1.
  6. Negative exponents: a−n = 1 / an for a ≠ 0
    Reason: use quotient rule with smaller exponent on top: a0 ÷ an = a0−n = a−n = 1/an.

4) Worked examples (showing each small step)

Example A: Simplify 23 · 24.

  1. Use product rule: add exponents → 23+4 = 27.
  2. Compute if needed: 27 = 128.

Example B: Simplify (32)4.

  1. Use power of a power: multiply exponents → 32·4 = 38.
  2. Compute: 38 = 6561 (only if asked).

Example C: Simplify 50 and 2−3.

  1. 50 = 1 (zero exponent rule).
  2. 2−3 = 1 / 23 = 1/8 (negative exponent rule).

Example D: Simplify (2·3)2 and compare with 22·32.

  1. (2·3)2 = (6)2 = 36.
  2. 22·32 = 4·9 = 36. They match (power of a product rule).

5) Common mistakes to avoid

  • Don’t add exponents when multiplying different bases: 23·32 is NOT 26 or 35; leave as 23·32 or compute numeric values.
  • Be careful with negative signs: (−2)3 = −8 but −23 usually means −(23) = −8 as well; however (−2)2 = +4 while −22 = −4 if written without parentheses. Use parentheses to be clear.
  • Follow order of operations — exponents come before multiplication and addition.

6) Practice problems (try these)

  1. Simplify: 42 · 4−1.
  2. Simplify: (−3)2 and −32 — explain the difference.
  3. Simplify: (23)2 · 2−4.
  4. Simplify: (1/2)3 and write as 2−3.

7) Solutions (check your work)

  1. 42 · 4−1 = 42+(−1) = 41 = 4.
  2. (−3)2 = 9 because the negative is inside the power and squared; −32 without parentheses is conventionally −(32) = −9. Parentheses matter.
  3. (23)2 · 2−4 = 23·2 · 2−4 = 26−4 = 22 = 4.
  4. (1/2)3 = 1/8, and 2−3 = 1/23 = 1/8 (they are the same).

8) Quick checklist when you see exponent problems

  • Are the bases the same? If yes, you may add/subtract exponents.
  • Are exponents nested (a power of a power)? Multiply exponents.
  • Are negative exponents present? Rewrite as reciprocals if helpful.
  • Are parentheses clear? If not, rewrite so the meaning is unambiguous.

Next steps

Tell me which chapter or problem you want next. I can:

  • Walk through a different chapter (fractions, ratios, geometry, etc.),
  • Help with homework problems from the book (give the chapter and problem number), or
  • Create a short practice set tailored to your weak spots and track progress.

If you want, send one specific problem now and I’ll solve it step-by-step like above.

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