Where this sits. This is one of the five algebra families — a little over half the test, and where most students have the least ground under them. Every family page on this site has the same eight parts, in the same order, so once you know the shape of one you know the shape of all of them. Where a part is empty, it says so.
DrillsSixteen drills for this family, one skill each, ten problems at a timeGo to the drills ↓Repeated multiplication and its undoing: squares and square roots, cubes and cube roots. Then the rules that let powers be combined — add the exponents when you multiply, subtract when you divide, multiply when you raise a power to a power — and what a zero, a negative, or a fractional exponent means.
Scientific notation is the same idea put to work on numbers too big or too small to write out. This family has the most material on the site of any of the fourteen: five quizzes, two apps, and a printable chart.
What the test asks in this family
- What a raised number countsQuiz 29
- Squares and cubesQuiz 29 · Squares, Roots, and Powers
- The perfect squares, by sightQuiz 29 · Squares, Roots, and Powers
- Powers of tenQuiz 29 · Quiz 32
- Exponents inside the order of operationsQuiz 29 · Quiz 6
- Square roots and cube rootsQuiz 30
- Roots that fall between whole numbersQuiz 30 · Squares, Roots, and Powers
- Multiplying powers: add the exponentsQuiz 31 · The Exponent Machine
- Dividing powers: subtract the exponentsQuiz 31 · The Exponent Machine
- A power of a power: multiply the exponentsQuiz 31 · The Exponent Machine
- Zero and negative exponentsQuiz 31 · The Rules, One Page
- Fractional exponentsQuiz 31a
- An exponent outside parenthesesQuiz 31a
- Writing a number in scientific notation, and reading it backQuiz 32
- Comparing numbers in scientific notationQuiz 32
- Multiplying and dividing in scientific notationQuiz 32
All 16 have something on this site.
Seven questions across this family. Type an answer and press Check. You get the reasoning for every one, right or wrong.
- 25 = 32. The raised number counts how many times the base is used as a factor, not what to multiply by. So this is 2 × 2 × 2 × 2 × 2 = 32, not 2 × 5 = 10. That misreading is the most common error in this family and it never stops being worth checking.
- √144 = 12, because 12 × 12 = 144. A square root undoes a square: it asks what number, multiplied by itself, gives this. The perfect squares up to 400 are worth knowing by sight, because the test leans on them and the calculator costs you time.
- Write as a single power: x4 · x3 x7. When you multiply powers of the same base, add the exponents. It is not a trick: x4 is four x's multiplied and x3 is three more, so altogether seven x's are being multiplied. Multiplying the exponents would give x12, which is a different and much larger thing.
- 50 = 1. Any number to the power zero is 1. Follow the ladder down and you can see why: 53 = 125, 52 = 25, 51 = 5 — each step divides by 5 — so the next step is 5 ÷ 5, which is 1. Not zero.
- (23)2 = 64. A power of a power multiplies the exponents: (23)2 = 26 = 64. You can check it the long way — 23 is 8, and 82 is 64. Two different rules live close together here: add when multiplying powers, multiply when raising a power to a power.
- 3−2 = 1⁄9. A negative exponent does not make the answer negative — it flips the number over. 3−2 = 1⁄3² = 1⁄9. Keep going down that same ladder past 30 = 1 and this is where it lands.
- Write 0.00042 in scientific notation. 4.2 × 10−4. Scientific notation puts one digit before the point and lets a power of ten carry the size. Move the point four places to the right to make 4.2, and because the original number is smaller than one, the exponent is negative. A number bigger than ten would take a positive exponent.
Missed one? Five quizzes cover this family in Learn it, The Exponent Machine works every rule one step at a time, and The Rules, One Page under On paper puts all eight on a sheet you can keep beside you.
Want more than these seven questions? Flyover I asks about all fourteen families in one sitting and ends in a grid that says where to start; Flyover II asks them again later, with different questions, so you can see what has moved.
- Quiz 29ExponentsWhat a power is, and what it is not.
- Quiz 30Square and Cube RootsUndoing a power, and the ones worth knowing by heart.
- Quiz 31Rules of ExponentsThe rules that let powers be combined.
- Quiz 31aFractional ExponentsA root written as a power, and why that is the same thing.
- Quiz 32Scientific NotationVery big and very small numbers, written short.
These five are Creatures in Flight, the fifth leg of the quiz series.
Sixteen short drills, one skill each, in order. Each starts with two worked examples, then gives you ten problems (or five) and the reasoning for every answer. Each one prints, with the answer key on the last page. When you get all ten right, or all five right twice in a row, the drill gets a check mark on the Drills page.
- 10.01An exponent as repeated multiplication
- 10.02The squares from 1 to 20
- 10.03Square roots of perfect squares
- 10.04Which two whole numbers is a square root between?
- 10.05Cubes and cube roots
- 10.06Powers of 10
- 10.07Zero and negative exponents
- 10.08Multiply powers with the same base
- 10.09Divide powers with the same base
- 10.10A power of a power, and a power of a product
- 10.11Fractional exponents: 1/2, 1/3, 3/2
- 10.12Simplify a square root (√50 = 5√2)
- 10.13Scientific notation to an ordinary number
- 10.14An ordinary number to scientific notation
- 10.15Multiply and divide in scientific notation · calculator allowed
- 10.16Compare numbers in scientific notation
- The Exponent Machine · works every rule one step at a time, and reads aloud if you want it to.
- Squares, Roots, and Powers · the twenty squares and their roots, powers of ten, the power chains and cubes, roots that fall between whole numbers, and the right-triangle numbers the test uses. Look, then practice on the keypad.
- The Rules, One Page · all eight rules on one sheet, made to print and keep beside you.
- Squares, Roots, and Powers · the same page, made to print: page two of the times table.
- Khan Academy, math · free videos and practice. Search it for exponents, square roots, exponent properties, scientific notation. These are the clips we start from in class.
- Math Antics · short, plain videos on the basics.
- Light & Salt Learning · free GED classes online, with playlists by topic.
- Get Sum Math · GED math on video, one topic at a time.
- Kaplan GED Test Prep and the McGraw-Hill Mathematical Reasoning Workbook · look up exponents, roots, and scientific notation in the contents of either.
On to expressions and polynomials. An expression is not a question; it is an amount with a variable still in it, and the exponent rules you have just learned are half of what you do to one.