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Grade 8 · unit review

Review: Integer Powers and Scientific Notation

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.

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Bring the unit skills together

Use these 4 new cases to check your reasoning. Choices are temporary and are not saved to an account, sent to staff or recorded as school grades. Download your current review choices before leaving or reloading.

Unit review questions
1. Simplify x^7·x^8.
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. Evaluate 2^(−7) exactly.
Need a hint?
  • A negative integer exponent represents the reciprocal of a positive power of the same nonzero base.
  • Keep a negative exponent separate from a negative base or leading minus sign.
3. Multiply (2×10^7)(3×10^8); give normalized scientific notation.
Need a hint?
  • Scientific-notation multiplication first multiplies the coefficients and then multiplies the powers of ten.
  • Normalized notation requires coefficient magnitude at least one and below ten for a positive value.
4. Add 3×10^7 and 4×10^6; give scientific notation.
Need a hint?
  • Adding scaled quantities requires equal scale factors.
  • Adding exponent values is not an addition rule.

The download contains your current choices and any checked feedback. Keep the separate transfer-task work in your own document; downloading its instructions does not mark it complete.

Use the skills in a new task

Create a four-part mathematical solution sheet for these supplied cases. 1. Simplify x^9·x^10. Candidate answers: (A) 2x^19 (B) x^1 (C) x^19 (D) x^90. 2. Evaluate 2^(−9) exactly. Candidate answers: (A) -1/512 (B) -512 (C) 1/512 (D) 512. 3. 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. 4. Add 3×10^9 and 4×10^8; give scientific notation. Candidate answers: (A) 3.4×10^8 (B) 3.4×10^9 (C) 7×10^17 (D) 7×10^9.

Work on paper or in your own document. Explain the choices you made.

  • State each requested quantity or relationship and its input restrictions.
  • Show intermediate calculations or the applicable theorem with matching correspondences.
  • Give all four results, including exact values or requested rounding and units.
  • Check a result by substitution, reversal, dimensions, a representation or a counterexample as appropriate.
Compare with a model response

1. x^19. Both factors have base x. Count the repeated x factors: 9+10=19. Multiplying same-base powers adds exponents. 2. 1/512. A negative exponent means reciprocal: 2^(−9)=1/(2^9)=1/512. It does not make the value negative. 3. 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. 4. 3.4×10^9. Rewrite 4×10^8 as 0.4×10^9. Add coefficients only after matching powers: (3+0.4)×10^9=3.4×10^9.

A different response can also meet the criteria. Compare the evidence and reasoning, rather than copying the wording.