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Grade 8 · lesson 1 of 4

Count Factors to Combine Same-Base Powers

Multiplying two powers with the same base concatenates their repeated factors.

About 20 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: Simplify x^2·x^3. Candidate answers: (A) 2x^5 (B) x^1 (C) x^5 (D) x^6.
  • Independent check inputs: Simplify x^3·x^4. Candidate answers: (A) 2x^7 (B) x^1 (C) x^12 (D) x^7.

Learn the idea

Powers encode repeated multiplication. Integer exponent rules preserve that structure, while scientific notation separates a number’s scale from a coefficient that can be compared or operated on efficiently. Multiplying two powers with the same base concatenates their repeated factors. Counting those factors explains why exponents add rather than multiply in a same-base product.

Here is the method on the supplied teaching case. Both factors have base x. Count the repeated x factors: 2+3=5. Multiplying same-base powers adds exponents.

The same-base condition is essential; these rules do not combine unrelated bases by adding exponents. Parentheses also distinguish a power of a product from a product of powers.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: Simplify x^2·x^3. Candidate answers: (A) 2x^5 (B) x^1 (C) x^5 (D) x^6. Solution: Both factors have base x. Count the repeated x factors: 2+3=5. Multiplying same-base powers adds exponents. The final answer is x^5. The same-base condition is essential; these rules do not combine unrelated bases by adding exponents. Parentheses also distinguish a power of a product from a product of powers.

Your turn

Check your reasoning

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Helpful tips
  • Multiplying two powers with the same base concatenates their repeated factors.
  • The same-base condition is essential; these rules do not combine unrelated bases by adding exponents.
Simplify x^3·x^4.

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See the answer and explanation

x^7

Both factors have base x. Count the repeated x factors: 3+4=7. Multiplying same-base powers adds exponents.

Take it further

Changed-input task: Simplify x^9·x^10. Candidate answers: (A) 2x^19 (B) x^1 (C) x^19 (D) x^90. Show the mathematical reason, not only the selected answer. Expected result: x^19. Both factors have base x. Count the repeated x factors: 9+10=19. Multiplying same-base powers adds exponents.

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More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. Simplify x^4·x^5.
Need a hint?
  • Multiplying two powers with the same base concatenates their repeated factors.
  • The same-base condition is essential; these rules do not combine unrelated bases by adding exponents.
2. Simplify x^5·x^6.
Need a hint?
  • Multiplying two powers with the same base concatenates their repeated factors.
  • The same-base condition is essential; these rules do not combine unrelated bases by adding exponents.
3. Simplify x^6·x^7.
Need a hint?
  • Multiplying two powers with the same base concatenates their repeated factors.
  • The same-base condition is essential; these rules do not combine unrelated bases by adding exponents.

Your choices are temporary. Feedback appears after you check all the answers.

Nothing here is saved to an account or sent to a teacher. Keep your own notes or download them before leaving. The score describes these questions; it does not certify subject mastery.

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These LifeStarter passages, scenarios and exercises are authored for this Academy with AI assistance and review. External resources keep their publishers’ terms. A grade label suggests a starting level; it does not establish alignment with your school’s complete requirements.