1.3 Exponents, Roots, and Scientific Notation

Exponents and Their Rules

Exponents, or powers, indicate repeated multiplication of a base number. In the expression 3^4, the base is 3 and the exponent is 4, meaning 3 × 3 × 3 × 3 = 81. To succeed on the ASTB-E without a calculator, you must master the fundamental rules of exponents.

The Multiplication Rule states that when multiplying terms with the same base, you add the exponents: x^a × x^b = x^(a+b). For example, 2^3 × 2^4 = 2^7.

The Division Rule dictates that when dividing terms with the same base, you subtract the exponents: x^a / x^b = x^(a-b). Thus, 5^6 / 5^2 = 5^4.

The Power to a Power Rule applies when an exponent is raised to another exponent; in this case, you multiply them: (x^a)^b = x^(a×b). Therefore, (3^2)^3 = 3^6.

Understanding special exponent cases is equally important. Any non-zero base raised to the power of zero is 1 (x^0 = 1). A negative exponent indicates the reciprocal of the base raised to the positive opposite of that exponent: x^-a = 1 / x^a. For instance, 4^-2 = 1 / 4^2 = 1/16.

Roots and Radicals

Roots are the inverse operation of exponents. The most common is the square root, denoted by √. The square root of a number x is the value that, when multiplied by itself, equals x. For example, √49 = 7. Higher roots, such as cube roots ∛, seek a number that, when multiplied by itself three times, yields the base (e.g., ∛64 = 4).

Roots can also be expressed as fractional exponents. The general rule is x^(a/b) = ^b√(x^a). For example, 8^(2/3) is the cube root of 8^2. It is often easier to take the root first: the cube root of 8 is 2, and 2^2 = 4. Memorizing perfect squares up to 15 and perfect cubes up to 5 will significantly accelerate your calculations on the exam:

Operations with Radicals

Simplifying radicals is a critical skill. A radical is in its simplest form when no perfect square factors remain inside the square root. To simplify √72, find the largest perfect square factor, which is 36. Rewrite it as √(36 × 2), which simplifies to 6√2.

You can only add or subtract radicals if they have the exact same number under the root (the radicand). For example, 3√5 + 2√5 = 5√5. However, you cannot directly add √2 and √3. Sometimes you must simplify first: √8 + √18 becomes 2√2 + 3√2 = 5√2.

Multiplying and dividing radicals allows you to combine the radicands. √a × √b = √(ab). For instance, √3 × √12 = √36 = 6. Division works similarly: (√a) / (√b) = √(a/b).

Rationalizing the denominator is the process of eliminating a radical from the bottom of a fraction. To rationalize 5 / √3, multiply the numerator and the denominator by √3 to get (5√3) / 3.

Scientific Notation

Scientific notation is a method of writing extremely large or microscopic numbers in a compact form: a × 10^n, where 1 ≤ a < 10 and n is an integer. This is incredibly useful for engineering and physics calculations found throughout the ASTB-E.

To convert 4,500,000 into scientific notation, place the decimal after the first non-zero digit to get 4.5, and count the number of places the decimal moved (6 places to the left). The result is 4.5 × 10^6. For very small numbers like 0.00032, move the decimal to the right to get 3.2, resulting in a negative exponent: 3.2 × 10^-4.

Calculating with Scientific Notation

When multiplying numbers in scientific notation, you multiply the coefficients (the a values) and add the exponents of the 10s. For example, (2 × 10^3) × (4 × 10^5) equals 8 × 10^8. If the resulting coefficient is 10 or greater, you must adjust it. (5 × 10^4) × (3 × 10^3) yields 15 × 10^7, which must be rewritten as 1.5 × 10^8.

Dividing follows a similar logic: divide the coefficients and subtract the exponents. (8 × 10^6) / (2 × 10^2) equals 4 × 10^4.

Adding or subtracting in scientific notation requires that the numbers have the exact same power of 10. To add 3 × 10^4 and 2 × 10^3, convert the latter to match the larger exponent: 0.2 × 10^4. Now you can add them to get 3.2 × 10^4. Mastery of these mechanics will allow you to quickly evaluate complex physics calculations without relying on prohibited calculators.