2.3 Squares, Square Roots, Cubes, Cube Roots, Undefined Expressions & Scientific Notation
Key Takeaways
- Q.2.b and Q.2.c test computing with and writing expressions using squares/square roots and cubes/cube roots of rational numbers — skills that resurface constantly in geometry (area, volume) and quadratic equations.
- Cube roots of negative numbers are defined and negative (the cube root of −27 is −3), but square roots of negative numbers are undefined within the real-number system tested on the GED.
- Q.2.d ('determine when a numerical expression is undefined') is most often tested as division by zero — any expression with a denominator that can equal zero is undefined at that value.
- Scientific notation writes a number as a × 10^n where 1 ≤ a < 10 — Q.2.e explicitly includes 'multi-step arithmetic problems involving scientific notation.'
- When multiplying or dividing numbers in scientific notation, combine the coefficients and add or subtract the exponents, then renormalize the result if the coefficient falls outside 1–10.
Why This Topic Matters
The remaining three indicators of assessment target Q.2 — Q.2.b (squares and square roots), Q.2.c (cubes and cube roots), and Q.2.d (when a numerical expression is undefined) — along with the scientific-notation half of Q.2.e, round out the arithmetic foundation the rest of the test depends on. Perfect squares and square roots reappear directly in geometry (area of a square, the Pythagorean theorem) and in solving quadratic equations later in the Algebraic domain (Chapter 9). Scientific notation shows up whenever a word problem involves very large numbers (population, distance in space, national budgets) or very small numbers (bacteria size, wavelengths) — common GED word-problem contexts.
Perfect Squares and Square Roots (Q.2.b)
Memorizing perfect squares through 15 and their square roots saves valuable time on both the no-calculator and calculator sections.
| n | n² | n | n² |
|---|---|---|---|
| 1 | 1 | 9 | 81 |
| 2 | 4 | 10 | 100 |
| 3 | 9 | 11 | 121 |
| 4 | 16 | 12 | 144 |
| 5 | 25 | 13 | 169 |
| 6 | 36 | 14 | 196 |
| 7 | 49 | 15 | 225 |
| 8 | 64 |
A square root asks "what number times itself gives this value?" So √81 = 9 because 9 × 9 = 81. The GED tests square roots of positive rational numbers only, including non-perfect squares approximated with a calculator (√50 ≈ 7.07) and square roots of fractions (√(4/9) = 2/3, since (2/3)² = 4/9).
Estimating a non-perfect-square root without a calculator (useful on the no-calculator section): find the two consecutive perfect squares the value falls between. √50 falls between √49 = 7 and √64 = 8, so √50 is between 7 and 8 — and since 50 is much closer to 49 than to 64, a reasonable estimate is about 7.1. This estimation skill previews the Pythagorean theorem work in Chapter 4, where square roots of non-perfect squares appear constantly when solving for a missing side length.
Perfect Cubes and Cube Roots (Q.2.c)
| n | n³ | n | n³ |
|---|---|---|---|
| 1 | 1 | 6 | 216 |
| 2 | 8 | 7 | 343 |
| 3 | 27 | 8 | 512 |
| 4 | 64 | 9 | 729 |
| 5 | 125 | 10 | 1,000 |
A cube root asks "what number times itself three times gives this value?" Unlike square roots, cube roots of negative numbers are defined and negative: the cube root of −27 is −3, because (−3) × (−3) × (−3) = −27. This is a key conceptual difference the GED tests directly — students who assume 'you can't take the root of a negative number' will miss cube-root items that have perfectly valid negative answers.
Q.2.d — Determining When an Expression Is Undefined
The GED tests this indicator most commonly as division by zero: any fraction or rational expression is undefined for any value that makes its denominator equal zero.
Worked example: For what value of x is the expression 12/(x − 5) undefined?
The expression is undefined when the denominator equals zero: x − 5 = 0, so x = 5. At x = 5, the expression 12/0 has no defined value.
A second, related idea connects back to Q.2.b: within the real-number system tested on the GED, the square root of a negative number is undefined (there is no real number that, multiplied by itself, produces a negative result) — even though the cube root of the same negative number is perfectly valid. Recognizing 'denominator = 0' or 'even root of a negative number' as the two situations that make an expression undefined covers the vast majority of Q.2.d items.
Scientific Notation (Part of Q.2.e)
Scientific notation expresses a number as a × 10^n, where the coefficient a satisfies 1 ≤ a < 10 and n is an integer (positive for large numbers, negative for small numbers).
| Standard Form | Scientific Notation |
|---|---|
| 45,000,000 | 4.5 × 10^7 |
| 0.00032 | 3.2 × 10^−4 |
| 6,200 | 6.2 × 10^3 |
Multiplying/dividing in scientific notation: multiply or divide the coefficients, and add (for multiplication) or subtract (for division) the exponents.
Worked example: (3 × 10^5) × (2 × 10^3) = (3 × 2) × 10^(5+3) = 6 × 10^8
Renormalizing trap: (5 × 10^4) × (4 × 10^3) = 20 × 10^7 — but 20 is not between 1 and 10, so this must be rewritten as 2.0 × 10^8 (moving the decimal one place shifts the exponent up by one). Forgetting to renormalize a coefficient of 10 or greater is the most common scientific-notation error on the GED.
Exam scenario: "A hard drive stores 2.4 × 10^12 bytes. If a photo file uses 6 × 10^6 bytes, how many photos can the drive store?" → (2.4 × 10^12) ÷ (6 × 10^6) = (2.4 ÷ 6) × 10^(12−6) = 0.4 × 10^6 = 4 × 10^5 photos (renormalized because 0.4 is less than 1).
What is the cube root of −125?
For what value of x is the expression 20/(x + 8) undefined?
Simplify and write in proper scientific notation: (8 × 10^5) × (3 × 10^2)
Key Takeaways
- Memorize perfect squares through 15² and perfect cubes through 10³ — recall speed on these pays off across geometry, quadratics, and Q.2 items alike.
- Cube roots of negative numbers are defined and negative; square roots of negative numbers are undefined in the real-number system the GED tests — know the difference.
- An expression is undefined whenever its denominator equals zero; solve 'denominator = 0' to find the excluded value.
- Scientific notation requires the coefficient to satisfy 1 ≤ a < 10 — always renormalize a result like 24 × 10^7 into 2.4 × 10^8.
- To multiply or divide numbers in scientific notation, combine the coefficients and add or subtract the exponents, then check the coefficient is still in proper range.