M2. Exponents and Scientific Notation

Exponents and Scientific Notation

Product Rule of Exponents

^{For any real number a and natural numbers m and n, the product rule of exponents states that}

a^m ⋅ aⁿ = a^m+n

Q. Write each of the following products with a single base.

Quotient Rule of Exponents

For any real number a and natural numbers m and n, such that m>n, the quotient rule of exponents states that

$\dpi{120} \frac{a^m}{a^n}=a^{m-n}$

Q. Write each of the following products with a single base.

Power Rule of Exponents

For any real number a and positive integers m and n, the power rule of exponents states that

$\dpi{120} \left ( a^m \right )^n=a^{m\cdot n}$

Q. Write each of the following products with a single base.

Zero Exponent Rule of Exponents

For any nonzero real number a, the zero exponent rule of exponents states that

a⁰ = 1

Q. Simplify each expression using the zero exponent rule of exponents.

Negative Rule of Exponents

For any nonzero real number a and natural number n, the negative rule of exponents states that

$\dpi{120} a^{-n}=\frac{1}{a^n}$

Q. Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.

Power of a Product Rule of Exponents

For any real numbers a and b and any integer n, the power of a product rule of exponents states that

(ab)ⁿ = aⁿ bⁿ

Q. Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.

Power of a Quotient Rule of Exponents

For any real numbers a and b and any integer n, the power of a quotient rule of exponents states that

$\dpi{120} \left (\frac{a}{b} \right )^n=\frac{a^n}{b^n}$

Q. Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.

Simplification of Exponential Expressions

To simplify an expression means to rewrite the given expression by combining terms or exponents; in other words, to write the expression more simply with fewer terms. The rules for exponents may be combined to simplify expressions.

Q. Simplify each expression and write the answer with positive exponents only.

Scientific Notation

A number is written in scientific notation if it is written in the form a × 10ⁿ, where 1≤∣a∣<10 and n is an integer.

Q. Write each number in scientific notation.

Converting from Scientific to Standard Notation

To convert a number in scientific notation to standard notation, simply reverse the process. Move the decimal n places to the right if n is positive or n places to the left if n is negative and add zeros as needed. Remember, if n is positive, the value of the number is greater than 1, and if n is negative, the value of the number is less than one.

Q. Convert each number in scientific notation to standard notation.