L5 · Exponents and scientific notation clinic

Learning goal

“I can choose the correct exponent law for any expression, evaluate powers with negative bases correctly, and move between standard form, powers of 10 and scientific notation without the unit's standard errors.”

The five moves — product: add exponents; quotient: subtract; power of a power: multiply; power of a product/quotient: distribute onto every factor.

Zero and negative exponents — a⁰ = 1 (a ≠ 0); a^(-n) = 1/a^n - reciprocal, never a negative value for a positive base.

Base — decided by parentheses: (-2)⁴ ≠ -2⁴.

Scientific notation — a × 10^n with 1 ≤ a < 10; conversion only in this course.

Worked example

Simplify (3²)² ÷ 3³. Step 1: Name the operations: a power of a power, then a division. Step 2: Apply in order: (3²)² = 3⁴ (multiply 2 × 2); 3⁴ ÷ 3³ = 3¹ (subtract 4 - 3). Step 3: Answer: 3. Cross-check by values: (9)² = 81; 81 ÷ 27 = 3. Say it plainly: the law follows the OPERATION, not the look of the expression. When in doubt, expand - the expansion is the judge.

Check yourself — explain it

In (x⁴)³ · x², which law fires first?

Paired problem — example still visible

Simplify (a⁴)³ ÷ a⁵ -> a^(12-5) = a⁷.

Faded completion — last step is yours

"2⁶ ÷ 2² = 2^___ = ___" -> 4; 16

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

Start practice →Take the unit quiz