6.3 Powers, Roots, and Scientific Notation
Key Takeaways
- Exponents show repeated multiplication, and parentheses determine whether a negative base is being raised to a power.
- Square roots and cube roots reverse powers, but square roots of positive numbers use the principal nonnegative root in GED numeric computation.
- Scientific notation writes a number as a coefficient from 1 up to but not including 10 multiplied by a power of 10.
- Multiplying or dividing numbers in scientific notation requires combining coefficients and powers of 10, then normalizing the result.
- A numerical expression can be undefined, especially when division by zero appears after simplifying or substituting a value.
Powers and Roots in GED Number Sense
The GED Mathematical Reasoning assessment includes powers, roots, and scientific notation because they help express growth, area, volume, and very large or very small quantities. These problems often look symbolic, but the main skill is reading the notation correctly and applying order of operations.
An exponent tells how many times to use a base as a factor. For example, 5^3 means 5 x 5 x 5 = 125. A root reverses a power: sqrt(144) = 12 because 12^2 = 144, and the cube root of 64 is 4 because 4^3 = 64.
Rules You Need Often
| Rule | Example | Result |
|---|---|---|
| Same base, multiply | 2^3 x 2^4 = 2^(3+4) | 2^7 |
| Same base, divide | 5^6 / 5^2 = 5^(6-2) | 5^4 |
| Power of a power | (3^2)^4 = 3^(2 x 4) | 3^8 |
| Zero exponent | 9^0 | 1 |
| Square root | sqrt(81) | 9 |
| Cube root | cube root of -27 | -3 |
These rules work when the base is the same. You cannot combine 2^3 and 5^3 by adding exponents, because the bases differ. If the bases are different but the exponents match, you may sometimes rewrite: 2^3 x 5^3 = (2 x 5)^3 = 10^3. That shortcut works because both factors are raised to the same power.
Parentheses and Negative Numbers
Parentheses can change the answer.
Worked example: Compare (-4)^2 and -4^2.
In (-4)^2, the base is -4, so (-4) x (-4) = 16. In -4^2, the exponent applies to 4 first, then the negative sign is applied: -(4 x 4) = -16. If the negative sign is part of the base, it must be inside parentheses.
Roots and Undefined Expressions
The GED may ask you to compute with roots or identify when an expression is undefined. The most important undefined case is division by zero. For example, 12/(5 - 5) is undefined because the denominator becomes 0. It is not 0, and it is not 12.
Worked example: Evaluate sqrt(49) + 2^3 - 18/(7 - 4).
sqrt(49) = 7. Next, 2^3 = 8. The denominator is 7 - 4 = 3, so 18/3 = 6. The value is 7 + 8 - 6 = 9.
When roots appear in geometry or measurement, keep the context in mind. A side length, distance, or radius cannot be negative in a real object, so the practical square-root answer is the positive value. Algebra may allow two solutions later, but numeric GED measurement contexts usually use the nonnegative root.
Scientific Notation
Scientific notation has the form a x 10^n, where a is at least 1 and less than 10. Move the decimal point to create the coefficient, then count moves.
- 430,000 = 4.3 x 10^5.
- 0.00072 = 7.2 x 10^-4.
Positive powers of 10 make numbers larger. Negative powers make numbers smaller.
Worked example: Multiply (4.5 x 10^6)(2 x 10^-3).
Multiply coefficients: 4.5 x 2 = 9. Combine powers: 10^6 x 10^-3 = 10^3. The result is 9 x 10^3, or 9,000.
Worked example: Add 3.2 x 10^5 and 7.5 x 10^4.
Rewrite 7.5 x 10^4 as 0.75 x 10^5. Then 3.2 x 10^5 + 0.75 x 10^5 = 3.95 x 10^5. For addition and subtraction, the powers must match before coefficients are combined.
Which value is equivalent to (2^3 x 2^4) / 2^2?
A machine produces 3.6 x 10^5 parts each month. How many parts does it produce in 8 months, written in scientific notation?