Multiply Coefficients and Combine Powers of Ten

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Multiply (2×10^2)(3×10^3); give normalized scientific notation. Candidate answers: (A) 5×10^5 (B) 6×10^1 (C) 6×10^5 (D) 6×10^6.
• Independent check inputs: Multiply (2×10^3)(3×10^4); give normalized scientific notation. Candidate answers: (A) 5×10^7 (B) 6×10^1 (C) 6×10^12 (D) 6×10^7.

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. Scientific-notation multiplication first multiplies the coefficients and then multiplies the powers of ten. If needed, shift a factor of ten to keep the final coefficient in normalized range.

Here is the method on the supplied teaching case. Multiply coefficients 2×3=6 and add exponents 2+3=5. The coefficient 6 already lies from 1 up to but excluding 10.

Normalized notation requires coefficient magnitude at least one and below ten for a positive value. Add exponents only for multiplication of equal bases; the coefficient still follows ordinary multiplication.

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.

Check your reasoning

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Take it further

Changed-input task: Multiply (2×10^9)(3×10^10); give normalized scientific notation. Candidate answers: (A) 5×10^19 (B) 6×10^1 (C) 6×10^19 (D) 6×10^90. Show the mathematical reason, not only the selected answer. Expected result: 6×10^19. Multiply coefficients 2×3=6 and add exponents 9+10=19. The coefficient 6 already lies from 1 up to but excluding 10.

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