6.3 Powers, Roots, and Scientific Notation

Powers and Roots in GED Number Sense

The GED Mathematical Reasoning test includes powers, roots, and scientific notation because they express growth, area, volume, and very large or very small quantities. These items look symbolic, but the core skill is reading the notation correctly and applying the order of operations.

An exponent tells how many times to use a base as a factor: 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.

Exponent Rules You Need Often

These rules require the same base. You cannot combine 2^3 and 5^3 by adding exponents because the bases differ. When bases differ but the exponents match, you may regroup instead: 2^3 x 5^3 = (2 x 5)^3 = 10^3 = 1000. That shortcut works only because both factors carry the same power.

Parentheses and Negative Numbers

Parentheses can flip the answer.

Worked example: Compare (-4)^2 and -4^2.

In (-4)^2 the base is -4, so (-4)(-4) = 16. In -4^2 the exponent binds to 4 first, then the negative is applied: -(4 x 4) = -16. If the negative sign is meant to be part of the base, it must sit inside parentheses. The GED writes this pairing precisely to test whether you read the grouping.

Roots and Undefined Expressions

The most important undefined case is division by zero. For example, 12 / (5 - 5) is undefined because the denominator is 0; it is not 0 and it is not 12. Watch for a denominator that collapses to zero after you substitute a value.

Worked example: Evaluate sqrt(49) + 2^3 - 18 / (7 - 4).

sqrt(49) = 7, 2^3 = 8, and the denominator 7 - 4 = 3 gives 18 / 3 = 6. So 7 + 8 - 6 = 9. Follow the order of operations: roots and powers first, then the division, then left-to-right addition and subtraction.

In geometry and measurement, a length, distance, or radius cannot be negative, so the practical square-root answer is the nonnegative value. Algebra may later allow two solutions, but numeric GED measurement uses the principal root.

Scientific Notation

Scientific notation has the form a x 10^n, where 1 <= a < 10. Slide the decimal to create the coefficient, then count the moves.

• 430,000 = 4.3 x 10^5 (decimal moved 5 places left, positive power).
• 0.00072 = 7.2 x 10^-4 (decimal moved 4 places right, negative power).

Positive powers of 10 make numbers larger; negative powers make them smaller.

Worked example: Multiply (4.5 x 10^6)(2 x 10^-3).

Coefficients: 4.5 x 2 = 9. Powers: 10^6 x 10^-3 = 10^(6-3) = 10^3. Result: 9 x 10^3 = 9000.

Worked example: Add 3.2 x 10^5 and 7.5 x 10^4.

For addition the powers must match, so 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. A frequent trap is adding 3.2 and 7.5 directly without aligning the exponents.

Estimating With Powers of 10

Scientific notation is most useful for judging size quickly. Because the power of 10 controls the magnitude, you can compare two huge or tiny numbers by their exponents first and their coefficients second. For instance, 8.1 x 10^4 is larger than 9.9 x 10^3 even though 9.9 > 8.1, because 10^4 is ten times 10^3. When two numbers share an exponent, only then does the coefficient decide.

Perfect Squares and Cubes Worth Memorizing

Recognizing common perfect squares and cubes lets you simplify roots without the calculator on Part I. The squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, and 144 cover most GED items, and the cubes 1, 8, 27, 64, and 125 cover the rest. When a number is not a perfect square, the root is irrational; sqrt(50) is between sqrt(49) = 7 and sqrt(64) = 8, so it is about 7.07. Knowing the nearest perfect squares lets you bracket an answer and eliminate choices.

Worked example: A square patio has an area of 169 square feet. What is the length of one side?

The side length is sqrt(169). Since 13 x 13 = 169, the side is 13 feet. Because a physical side cannot be negative, the practical answer uses only the principal root of 13, not -13. This is exactly the kind of measurement item where the GED expects the nonnegative value, even though the equation x^2 = 169 technically has the two solutions 13 and -13.