6.1 Exponents, radicals & scientific notation
Key Takeaways
- Multiply like bases by adding exponents (a^m . a^n = a^(m+n)) and divide by subtracting them; never multiply the bases themselves.
- The zero exponent gives 1 (a^0 = 1) and a negative exponent means the reciprocal (a^(-n) = 1/a^n).
- A fractional exponent is a root: a^(1/n) is the nth root, and a^(m/n) means take the nth root and raise it to the m power.
- Scientific notation writes numbers as a x 10^n with 1 <= a < 10; large numbers use positive exponents and small numbers use negative ones.
- To multiply in scientific notation, multiply the coefficients and add the exponents; to divide, divide the coefficients and subtract the exponents, then re-normalize so the leading number is between 1 and 10.
Understanding Exponents
An exponent is shorthand for repeated multiplication. In 5^3, the number 5 is the base and 3 is the exponent (or power), so 5^3 = 5 x 5 x 5 = 125. On the PERT Mathematics subtest you are rarely asked to simply expand a power; instead you combine powers using the laws of exponents, which let you rewrite an expression quickly without multiplying everything out. Memorizing these five rules turns slow arithmetic into fast pattern-matching, and it saves precious time on the 30-item adaptive test.
The five laws of exponents
| Law | Rule | Example |
|---|---|---|
| Product | a^m . a^n = a^(m+n) | 2^3 . 2^4 = 2^7 = 128 |
| Quotient | a^m / a^n = a^(m-n) | x^7 / x^2 = x^5 |
| Power of a power | (a^m)^n = a^(m*n) | (3^2)^3 = 3^6 = 729 |
| Zero exponent | a^0 = 1 (a is not 0) | 17^0 = 1 |
| Negative exponent | a^(-n) = 1 / a^n | 2^(-3) = 1/8 |
The single most common mistake is multiplying the bases when you should add the exponents. Notice that 2^3 . 2^4 is not 4^7; the base stays 2 and only the exponents combine, giving 2^(3+4) = 2^7. Likewise, do not confuse the zero rule with zero itself: any nonzero number to the zero power equals 1, not 0.
Worked example 1. Simplify (6x^5 . x^2) / (3x^3). First combine the numerator with the product rule: x^5 . x^2 = x^(5+2) = x^7, so the numerator is 6x^7. Now divide: (6x^7)/(3x^3) = (6/3) . x^(7-3) = 2x^4. The coefficients divide normally, and the exponents subtract.
Worked example 2. Simplify (2a^3)^4. The power-of-a-power rule applies to every factor inside the parentheses, including the coefficient: (2a^3)^4 = 2^4 . a^(3*4) = 16a^12.
Worked example 2b. Rewrite 5x^(-2) using a positive exponent. A negative exponent sends its base to the denominator: 5x^(-2) = 5 / x^2. Only the x moves, because the exponent -2 attaches to x alone, not to the coefficient 5. This is why a negative exponent never makes a number negative -- it makes a fraction.
Roots and Radicals
A radical undoes an exponent. The square root of a asks "what number squared gives a?" Because 7 x 7 = 49, the square root of 49 is 7. A cube root asks what number cubed gives the value, so the cube root of 27 is 3 because 3^3 = 27.
Every radical can be rewritten as a fractional exponent, and this is the key idea the PERT tests:
- the square root of a = a^(1/2)
- the cube root of a = a^(1/3)
- the nth root of a = a^(1/n)
- a^(m/n) = the nth root of (a^m) = (the nth root of a)^m
The denominator of the fraction is the root; the numerator is the power. So 8^(2/3) means "cube-root 8, then square it": the cube root of 8 is 2, and 2^2 = 4. You could also square first (8^2 = 64) and then take the cube root (cube root of 64 is 4). The answer is the same, but rooting first keeps the numbers small and easier to handle mentally.
Worked example 3. Evaluate 16^(3/4). The denominator 4 says take the fourth root: the fourth root of 16 is 2, since 2^4 = 16. The numerator 3 says raise that to the third power: 2^3 = 8. So 16^(3/4) = 8.
To simplify a radical, pull out perfect-square factors. For the square root of 72, factor 72 = 36 x 2. Since 36 is a perfect square, the square root of 72 = (square root of 36)(square root of 2) = 6 times the square root of 2.
Scientific Notation
Scientific notation writes a number as a value between 1 and 10 multiplied by a power of 10. It keeps very large and very small numbers manageable. The form is a x 10^n, where 1 is less than or equal to a, and a is less than 10.
- Large numbers use a positive exponent: 47,000 = 4.7 x 10^4 (the decimal moved 4 places left).
- Small numbers use a negative exponent: 0.0035 = 3.5 x 10^(-3) (the decimal moved 3 places right).
To convert back to standard form, move the decimal the number of places the exponent indicates: 6.02 x 10^5 = 602,000 (move 5 right); 9 x 10^(-4) = 0.0009 (move 4 left).
Multiplying and dividing in scientific notation uses the exponent laws directly. Multiply the front numbers and add the exponents; divide the front numbers and subtract the exponents.
Worked example 4. Multiply (3 x 10^4)(2 x 10^5). Multiply the coefficients: 3 x 2 = 6. Add the exponents: 10^(4+5) = 10^9. The result is 6 x 10^9.
Worked example 5. Divide (8 x 10^7) / (4 x 10^3). Divide the coefficients: 8 / 4 = 2. Subtract the exponents: 10^(7-3) = 10^4. The result is 2 x 10^4.
Sometimes the front number lands outside the 1-to-10 range and you must re-normalize. For (5 x 10^3)(4 x 10^2), the coefficients give 20 and the exponents give 10^5, so you have 20 x 10^5. Because 20 = 2 x 10^1, rewrite the whole thing as 2 x 10^6. Always confirm that the leading number sits between 1 and 10 before you choose your answer.
Keep in mind that the exponent laws and scientific notation are the same tool: a power of 10 is just a special base, so every rule in the table above applies directly to it.
Using the product rule, what is 2^3 . 2^4?
Evaluate 8^(2/3).
Compute (6 x 10^8) / (3 x 10^2) and express it in scientific notation.