Exponents, Roots, and Scientific Notation

1.4: Exponents, Roots, and Scientific Notation. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 4 of 53 parts Practice Flashcards

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Integer exponents

Objective: Evaluate powers and zero exponents.

Prerequisite skills: multiplication and division, integer powers. An exponent counts repeated factors. A nonzero base to the zero power is 1. A negative exponent means the reciprocal of the corresponding positive power; it does not make a positive base negative. Common error: interpreting a negative exponent as a negative value. Restrictions and interpretation: Use principal nonnegative square roots; scientific notation requires 1 ≤ |a| < 10.

Formulas

$a^(−k) = 1/a^k, a ≠ 0$

Worked examples

Try itEvaluate 2³ × 2^(−22).Show answer

Combine equal bases by adding exponents: 2^(3 − 22) = 2^-19 = 0.00000191. Therefore the requested answer is 0.00000191.

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Integer exponents

Objective: Evaluate powers and zero exponents.

Formulas

$a^(−k) = 1/a^k, a ≠ 0$

Worked examples

Try itEvaluate 2³ × 2^(−22).Show answer

Combine equal bases by adding exponents: 2^(3 − 22) = 2^-19 = 0.00000191. Therefore the requested answer is 0.00000191.

Roots and powers

Objective: Recognize the principal square root and square-root equations.

The symbol √a denotes the nonnegative square root for a ≥ 0. Solving x² = a instead gives both positive and negative roots when a > 0. Cube roots can be negative. Common error: writing a negative value for a principal square root. Restrictions and interpretation: Use principal nonnegative square roots; scientific notation requires 1 ≤ |a| < 10.

Formulas

$√(u²) = |u|$

Worked examples

Try itEvaluate √((-25)²).Show answer

The square is 625. Its principal square root is nonnegative, so √625 = 25. Equivalently √(u²) = |u|. Therefore the requested answer is 25.

Scientific notation

Objective: Multiply and normalize numbers in scientific notation.

Multiply coefficients and add powers of ten. If the resulting coefficient is at least 10, move its decimal left and increase the exponent. Scientific notation helps keep magnitude explicit. Common error: multiplying the exponents instead of adding them. Restrictions and interpretation: Use principal nonnegative square roots; scientific notation requires 1 ≤ |a| < 10. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

$(a × 10^m)(b × 10^n) = ab × 10^(m+n)$

Worked examples

Try itWhat is the value of (25 × 10³)(3 × 10¹)?Show answer

Multiply coefficients: 25 × 3 = 75. Add exponents: 3 + 1 = 4. The value is 75 × 10⁴ = 750000. Normalize the coefficient if needed without changing the value. Therefore the requested answer is 750000.

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