Integers, Exponents & Roots

1 · Numbers below zero

On a number line, numbers to the left of 0 are negative. The farther left, the smaller: −9 is less than −2, even though 9 is more than 2.

Temperatures, elevations below sea level, and money owed are all everyday negatives. A diver at −30 feet is deeper than a diver at −12 feet.

2 · Adding and subtracting integers

Adding a negative moves left: 6 + (−9) = −3. Subtracting a negative moves right, because subtracting is adding the opposite: 4 − (−7) = 4 + 7 = 11.

A distance or a change is always "larger minus smaller". From −8°F up to 5°F is 5 − (−8) = 13 degrees. Adding 8 and 5 is the same idea: 8 degrees to reach 0, then 5 more.

3 · Multiplying and dividing with signs

Same signs give a positive; different signs give a negative. (−6) × (−4) = 24, (−6) × 4 = −24, and −36 ÷ 9 = −4.

With several factors, count the negatives. An even number of them gives a positive answer; an odd number gives a negative one. (−1) × (−2) × (−3) = −6.

4 · Absolute value

The absolute value of a number is its distance from 0, so it is never negative: |−15| = 15 and |15| = 15.

Work inside the bars first: |3 − 10| = |−7| = 7. The bars are not a way to "drop the minus signs" before you calculate.

5 · Exponents

An exponent says how many times to multiply a number by itself: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32, not 2 × 5.

Watch the parentheses. (−3)² = (−3) × (−3) = 9, but −3² means −(3 × 3) = −9: the exponent happens before the minus sign.

6 · Exponent rules

Same base, multiplying: add the exponents. 5³ × 5⁴ = 5⁷. Same base, dividing: subtract them. 2⁹ ÷ 2⁶ = 2³ = 8.

A power of a power multiplies the exponents: (3²)³ = 3⁶. Any number (except 0) to the power 0 is 1.

7 · Square roots and cube roots

√49 = 7 because 7 × 7 = 49. The cube root of 64 is 4 because 4 × 4 × 4 = 64, so a cube with a volume of 64 cubic inches has edges 4 inches long.

To estimate √70, find the perfect squares on each side: 8² = 64 and 9² = 81. So √70 is between 8 and 9, a little closer to 8.

8 · Scientific notation

Very big and very small numbers are written as a number from 1 to 10 times a power of 10. 4,500,000 = 4.5 × 10⁶, because the decimal point moves 6 places.

A negative exponent means a small number, not a negative one: 3.2 × 10⁻⁴ = 0.00032. To compare two numbers in scientific notation, look at the exponents first.

9 · Prime factorization, GCF, and LCM

Break a number into primes with a factor tree: 180 = 2 × 2 × 3 × 3 × 5 = 2² × 3² × 5.

Use the GCF when you split things into equal groups with nothing left over: 24 pencils and 36 erasers make at most 12 identical kits. Use the LCM when things repeat and you want them to line up again: lights blinking every 6 and every 8 seconds blink together every 24 seconds.

10 · Order of operations

Parentheses, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right.

20 − 2 × 3² = 20 − 2 × 9 = 20 − 18 = 2. Doing 20 − 2 first gives 162, a very different answer.

11 · On the ISEE

Many wrong choices come from one sign slip. Before you choose, ask whether the answer should be positive or negative, and whether it should be bigger or smaller than the numbers you started with.