Study guide: 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.
Ideas to explain in your own words
- 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. The same-base condition is essential; these rules do not combine unrelated bases by adding exponents.
- A negative integer exponent represents the reciprocal of a positive power of the same nonzero base. This extends the quotient pattern without making a positive base’s value negative. Keep a negative exponent separate from a negative base or leading minus sign.
- 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. Normalized notation requires coefficient magnitude at least one and below ten for a positive value.
- Adding scaled quantities requires equal scale factors. Rewrite one term with the other’s power of ten, then combine coefficients while preserving that shared power of ten. Adding exponent values is not an addition rule.
Useful vocabulary
- integer exponent
- A whole-number or negative whole-number power.
- reciprocal
- One divided by a nonzero value.
- scientific notation
- For these positive values, a coefficient at least 1 and less than 10 multiplied by an integer power of ten.
Watch your reasoning
- 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.
- Keep a negative exponent separate from a negative base or leading minus sign. Zero cannot be used as the base of a negative power because the reciprocal would require division by zero.
- 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.
A worked review example
Complete supplied guide examples: Count Factors to Combine Same-Base Powers — Simplify x^2·x^3. Candidate answers: (A) 2x^5 (B) x^1 (C) x^5 (D) x^6. Both factors have base x. Count the repeated x factors: 2+3=5. Multiplying same-base powers adds exponents. Interpret a Negative Exponent as a Reciprocal — Evaluate 2^(−2) exactly. Candidate answers: (A) -1/4 (B) -4 (C) 1/4 (D) 4. A negative exponent means reciprocal: 2^(−2)=1/(2^2)=1/4. It does not make the value negative. Multiply Coefficients and Combine Powers of Ten — 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. Multiply coefficients 2×3=6 and add exponents 2+3=5. The coefficient 6 already lies from 1 up to but excluding 10. Match the Power of Ten before Adding — Add 3×10^2 and 4×10^1; give scientific notation. Candidate answers: (A) 3.4×10^1 (B) 3.4×10^2 (C) 7×10^2 (D) 7×10^3. Rewrite 4×10^1 as 0.4×10^2. Add coefficients only after matching powers: (3+0.4)×10^2=3.4×10^2.
Make a practice plan
- Study the four methods in the listed order; attempt each independent check before opening its explanation.
- Complete each lesson’s three changed-input questions and record the operation or reasoning step behind any correction.
- Take the four-question unit review, then solve the transfer task with written justifications.
- After correcting errors, attempt the separate unit exam and revisit the linked lesson for any unresolved method.