The core idea
An exponent records repeated multiplication for positive whole-number powers. Consistent extension gives zero and negative exponents, while roots reverse powers. Scientific notation separates a number into a manageable coefficient and a power of ten, preserving both its size and its units.
1. A power counts repeated factors
In 5³, the number 5 is the base and the raised 3 is the exponent. It means 5 × 5 × 5 = 125, not 5 × 3. In plain text, the same power may be written 5^3; the caret introduces the exponent. A square such as 7² has two equal factors, giving 49. A cube such as 4³ has three equal factors, giving 64.
Brackets define the base when a negative sign is involved. The expression (−3)² means (−3) × (−3) = 9. Without brackets, −3² conventionally means the negative of 3², so it equals −9. In (−3)³ the three negative factors give −27. Read the grouped base before counting factors; otherwise a correct multiplication rule can be applied to the wrong number.
Sources: OpenStax: Prealgebra 2e: Multiplication Properties of Exponents ↗
2. Derive rules by tracking the factors
Multiplying powers with the same base combines their factors. For example, 2³ × 2⁴ has seven factors of 2, giving 2⁷ = 128. In general, a^m × a^n = a^(m+n) when these powers are defined. The letters m and n denote exponents, and the parentheses keep their sum together. The rule requires a common base; it does not directly combine 2³ × 3⁴.
For a nonzero base, division cancels common factors: 7⁵/7² = 7³ = 343. Raising a power to another power multiplies the exponents: (2³)² = 2⁶ = 64. Addition is different: 2³ + 2² = 8 + 4 = 12, not 2⁵. Expand a short example into factors whenever an exponent rule feels uncertain; the underlying multiplication is the reason the shortcut works.
Sources: OpenStax: Prealgebra 2e: Multiplication Properties of Exponents ↗ · OpenStax: Prealgebra 2e: Divide Monomials ↗
3. Zero and negative exponents extend the pattern
Start with 10³ = 1000, 10² = 100 and 10¹ = 10. Reducing the exponent by one divides the value by 10. Continuing gives 10⁰ = 1, 10⁻¹ = 0.1, 10⁻² = 0.01 and 10⁻³ = 0.001. A negative exponent signals a reciprocal, not a negative value. For any nonzero base a, a⁰ = 1 and a^(−n) = 1/a^n.
For example, 2⁻³ = 1/8, while −2³ = −8. The nonzero condition matters: 0⁻¹ would require division by zero. This lesson leaves 0⁰ undefined rather than applying the nonzero-base rule to it. A powers-of-ten table also helps check direction: multiplying by 10² makes a positive number one hundred times larger; multiplying by 10⁻² makes it one hundredth as large.
Powers of ten change place value
| Power | Value | Example |
|---|---|---|
| 10³ | 1,000 | 4.2 × 10³ = 4,200 |
| 10⁰ | 1 | 4.2 × 10⁰ = 4.2 |
| 10⁻³ | 0.001 | 4.2 × 10⁻³ = 0.0042 |
Sources: NCERT Class VIII Mathematics Exemplar: Exponents and Powers ↗ · OpenStax: Prealgebra 2e: Integer Exponents and Scientific Notation ↗
4. Roots reverse powers with a stated convention
The square-root symbol √ asks for the nonnegative number whose square is the given nonnegative value. Thus √81 = 9 because 9² = 81. But the equation x² = 81 has two real solutions, 9 and −9, since both square to 81. The symbol √81 itself denotes only 9. Distinguish evaluating a root from solving an equation.
For an illustrative square with area 2.25 square metres, side length is √2.25 = 1.5 metres; negative length is excluded by the context. Not every root is a terminating decimal. Since 4² = 16 and 5² = 25, √20 lies between 4 and 5. A cube root reverses cubing: ∛(−8) = −2 because (−2)³ = −8. The small 3 on the root symbol identifies a cube root.
Root operations do not distribute over addition. For example, √(9 + 16) = √25 = 5, while √9 + √16 = 3 + 4 = 7. The bracketed sum must be found before taking its root. Likewise, taking the square root after squaring a negative number does not return its negative sign: √((−6)²) = √36 = 6. Squaring loses the sign, and the square-root symbol chooses the nonnegative result. Test proposed shortcuts against these small examples before using them with complicated numbers.
Sources: NCERT Class VIII Mathematics Exemplar: Square–Square Root and Cube–Cube Root ↗ · OpenStax: Prealgebra 2e: Simplify and Use Square Roots ↗
5. Solved example: write large and small measurements
For a positive number, standard scientific notation has the form a × 10^n, with coefficient a at least 1 and less than 10, and integer exponent n. For example, 4,600,000 metres = 4.6 × 10⁶ metres. The coefficient gives the leading digits, while the power gives the scale. The expression 46 × 10⁵ has the same value but is not in this standard form because 46 exceeds 10.
An illustrative paper sheet is 0.00008 metre thick. Write it as 8 × 10⁻⁵ metre because 10⁻⁵ = 0.00001 and eight such units make 0.00008. Check by multiplying back, not merely counting zeros. A negative exponent here describes a small positive length. Zero is ordinarily written as 0; it cannot meet the nonzero coefficient condition of this standard form.
Sources: OpenStax: Prealgebra 2e: Integer Exponents and Scientific Notation ↗
6. Solved example: calculate a stack thickness
Stack 2500 of the illustrative sheets, assuming no gaps or compression. Write the count as 2.5 × 10³. Total thickness is (2.5 × 10³) × (8 × 10⁻⁵) metres. Multiply coefficients to get 20 and add exponents to get −2, giving 20 × 10⁻² = 0.20 metre. In standard form this is 2 × 10⁻¹ metre, equivalent to 20 centimetres.
Check in another unit: 0.00008 metre equals 0.08 millimetre, so 2500 × 0.08 = 200 millimetres, again 20 centimetres. In division, divide coefficients and subtract exponents. For example, 6 × 10⁵ metres divided by 2 × 10² seconds gives 3 × 10³ metres per second. This illustrative calculation shows how units travel with the arithmetic; it is not a measured speed.
Sources: OpenStax: Prealgebra 2e: Integer Exponents and Scientific Notation ↗ · OpenStax: Prealgebra 2e: Divide Monomials ↗
7. Align scales before adding or comparing
To add 4.2 × 10⁵ and 7 × 10⁴, rewrite the second as 0.7 × 10⁵. The sum is (4.2 + 0.7) × 10⁵ = 4.9 × 10⁵. Adding exponents would be a multiplication rule used in the wrong operation. To compare 3 × 10⁻⁴ and 8 × 10⁻⁵, rewrite the first as 30 × 10⁻⁵; it is larger even though its coefficient initially looks smaller.
Scientific notation does not create measurement precision. A rough length estimate remains rough after rewriting it with powers of ten. Keep the original units and any stated uncertainty, and avoid adding unsupported decimal digits. Before accepting a result, compare it with an order-of-magnitude estimate: the stack contains thousands of very thin sheets, so a length of hundreds of metres would signal an exponent error. This quick check is especially useful before reusing a result in another calculation.
Sources: OpenStax: Prealgebra 2e: Integer Exponents and Scientific Notation ↗
PUT IT INTO PRACTICE
Calculate a model storage total
- An illustrative digital file occupies 3.2 × 10⁶ bytes; a byte is a unit of stored information. Write 75 files as 7.5 × 10¹ files and set up the total-size multiplication.
- Multiply coefficients and combine powers, then rewrite the answer in standard scientific notation. Check by multiplying 3,200,000 by 75.
- Check 24 × 10⁷ = 2.4 × 10⁸ = 240,000,000 bytes. State the assumptions: identical file sizes and no additional storage overhead in this model.
Check your understanding
Simplify 3² × 3⁴ and explain the exponent.
It is 3⁶ = 729 because the product contains six factors of 3.
Is 5⁻² equal to −25?
No. It equals 1/5² = 1/25 = 0.04; the negative exponent specifies a reciprocal.
Find √49 and all real solutions of x² = 49.
√49 = 7, while the equation has solutions x = 7 and x = −7.
Write 0.0062 in standard scientific notation.
6.2 × 10⁻³. Multiplying 6.2 by 0.001 recovers the original value.
Add 2 × 10³ and 6 × 10² without confusing addition with multiplication.
2000 + 600 = 2600 = 2.6 × 10³. Rewrite 6 × 10² as 0.6 × 10³ before adding coefficients.