Click image to open full size
Exponents & Scientific Notation
Laws of Exponents, Powers of Ten, and Converting Large Numbers
Related Tools
Related Labs
Exponents are a compact way to show repeated multiplication, and scientific notation uses exponents to write very large or very small numbers efficiently. These ideas appear in algebra, physics, chemistry, computer science, and everyday calculations involving scale. Learning the rules of exponents helps students simplify expressions quickly and avoid long arithmetic. Scientific notation also makes it easier to compare quantities like the size of atoms or the distance to stars.
An exponent tells how many times a base is multiplied by itself, such as 2^4 = 2 x 2 x 2 x 2. The exponent rules let us combine, divide, and raise powers without expanding every factor. Scientific notation writes a number as a x 10^n, where 1 <= a < 10 and n is an integer. Positive exponents move the decimal to the right in value, while negative exponents represent numbers less than 1.
Key Facts
- a^m x a^n = a^(m+n)
- a^m / a^n = a^(m-n), for a != 0
- (a^m)^n = a^(mn)
- a^0 = 1, for a != 0
- a^(-n) = 1 / a^n
- Scientific notation: N = a x 10^n, where 1 <= a < 10
Vocabulary
- Base
- The base is the number or variable that is being multiplied repeatedly in an exponential expression.
- Exponent
- The exponent tells how many times the base is used as a factor.
- Power
- A power is an expression made of a base and an exponent, such as 5^3.
- Scientific notation
- Scientific notation is a way to write a number as a value between 1 and 10 multiplied by a power of 10.
- Negative exponent
- A negative exponent means take the reciprocal of the base raised to the corresponding positive exponent.
Common Mistakes to Avoid
- Adding exponents when multiplying different bases, such as saying 2^3 x 3^3 = 6^6. This is wrong because the product rule only works when the bases are the same.
- Thinking a^0 = 0. This is wrong because any nonzero base raised to the zero power equals 1.
- Moving the decimal the wrong way in scientific notation. This is wrong because positive powers of 10 make the number larger and negative powers make it smaller.
- Forgetting to distribute an exponent to every factor inside parentheses, such as saying (2x)^3 = 2x^3. This is wrong because the exponent applies to both factors, so (2x)^3 = 2^3 x^3 = 8x^3.
Practice Questions
- 1 Simplify: (3^4 x 3^2) / 3^3.
- 2 Write 0.00056 in scientific notation, and write 4.2 x 10^5 in standard form.
- 3 A student says that 10^6 + 10^6 = 10^12 because the exponents should add. Explain why this reasoning is incorrect and give the correct result.