Why it matters on the HSPT
Exponents appear as a direct question, as a step inside a longer one, and as a comparison trap. The comparison items in Quantitative Skills love to put 2 to the 4th next to 4 squared next to 8, because a student who reads exponents as multiplication will get them all wrong the same way. Square roots turn up in geometry, in number series that alternate squares and roots, and in the Pythagorean rule. Scientific notation is a small topic, one or two items, but it is pure rule-following, so it is free points for anyone who knows the rule.
The rules
What an exponent means. The small raised number tells you how many times to multiply the base by itself. Five squared is 5 times 5, which is 25. Two cubed is 2 times 2 times 2, which is 8. Two to the 5th is 32. The exponent is a count of factors, never a number to multiply by.
Special cases. Anything to the first power is itself. Anything except zero to the zero power is 1. One to any power is 1. A negative base with an even exponent gives a positive; with an odd exponent, a negative. Negative 2 squared in parentheses is 4, and negative 2 cubed is negative 8.
Squares to know cold. The squares of 1 through 15 and the cubes of 1 through 5. Time spent memorizing them pays back on every section of the test.
Square roots. The square root of a number is the number that, squared, gives it back. The root of 49 is 7 because 7 squared is 49. On the HSPT, roots are of perfect squares, so if you know the squares, you know the roots. The root of 144 is 12. For a root that is not perfect, the test only asks you to place it between two whole numbers: the root of 50 is between 7 and 8, because 49 and 64 surround it.
Multiplying and dividing powers. With the same base, multiplying adds the exponents and dividing subtracts them. Two cubed times 2 squared is 2 to the 5th. Two to the 6th divided by 2 squared is 2 to the 4th. This appears occasionally; when in doubt, compute both numbers and check.
Powers of ten. Ten squared is 100, 10 cubed is 1,000, and in general the exponent counts the zeros. Multiplying by 10 to the n moves the decimal point n places right; dividing moves it left.
Scientific notation. A number is written as a value between 1 and 10 times a power of ten. Thirty-two thousand is 3.2 times 10 to the 4th. To read it, move the decimal point the number of places the exponent says, right for positive and left for negative. To write it, move the point until one nonzero digit is left of it and count how far you moved. A number with a bigger exponent is always larger, so compare exponents first and the front numbers only if the exponents tie.
Worked examples
Example 1
2⁴ =
- A.6
- B.8
- C.16
- D.32
Show answer
Answer: C: 16
Two used as a factor four times: 2 times 2 is 4, times 2 is 8, times 2 is 16. Choice B, 8, is 2 times 4, the error of reading the exponent as a multiplier.
Example 2
3² + 4² =
- A.14
- B.25
- C.49
- D.81
Show answer
Answer: B: 25
Nine plus 16 is 25. Choice C, 49, is 7 squared, the error of adding first and squaring after; the square applies to each number separately.
Example 3
√81 + √16 =
- A.9.5
- B.13
- C.25
- D.97
Show answer
Answer: B: 13
The root of 81 is 9 and the root of 16 is 4, so the sum is 13. Choice D, 97, adds inside the roots first, which is not allowed; roots are taken separately.
Example 4
Between which two whole numbers is √70?
- A.6 and 7
- B.7 and 8
- C.8 and 9
- D.34 and 36
Show answer
Answer: C: 8 and 9
Sixty-four is 8 squared and 81 is 9 squared, and 70 sits between them, so the root of 70 is between 8 and 9. Choice B is the trap for stopping at 49 and 64.
Example 5
(−3)² − (−2)³ =
- A.1
- B.5
- C.17
- D.−17
Show answer
Answer: C: 17
Negative 3 squared is 9, because an even exponent makes a positive. Negative 2 cubed is negative 8, because an odd exponent keeps the negative. Nine minus negative 8 is 9 plus 8, which is 17. Choice A, 1, comes from getting negative 2 cubed as positive 8.
Example 6
Which of the following equals 4,500?
- A.4.5 × 10²
- B.4.5 × 10³
- C.45 × 10³
- D.0.45 × 10²
Show answer
Answer: B: 4.5 × 10³
Four point five times 1,000 is 4,500. Choice A gives 450, choice C gives 45,000, and choice D gives 45. Count the places the decimal moves: from 4.5 to 4,500 is three places, so the exponent is 3.
Example 7
Which of the following is the largest?
- A.2⁵
- B.5²
- C.3³
- D.4² + 4
Show answer
Answer: A: 2⁵
Compute each: 2 to the 5th is 32, 5 squared is 25, 3 cubed is 27, and 4 squared plus 4 is 20. The largest is 32. Notice that 2 to the 5th and 5 squared are not the same number; the base and the exponent cannot be swapped.
Common mistakes
- Multiplying the base by the exponent. Two to the 4th is 16, not 8.
- Squaring a sum instead of each term. Three squared plus 4 squared is 25, not 49.
- Adding inside a root. The root of 9 plus the root of 16 is 7, not 5.
- Losing the sign on a negative base. Even exponent, positive; odd exponent, negative.
- Comparing scientific notation by the front number. Compare the powers of ten first.
Practice
Exponents and roots sit inside pre-algebra and arithmetic items, drive many non-geometric comparison traps, and show up in number series built on squares. The integers lesson covers the sign rules used here.