 # Exponents and Scientific Notation P.2. Definition of a Natural Number Exponent If b is a real number and n is a natural number, b n is read “the nth power.

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Exponents and Scientific Notation P.2

Definition of a Natural Number Exponent If b is a real number and n is a natural number, b n is read “the nth power of b” or “ b to the nth power.” Thus, the nth power of b is defined as the product of n factors of b. Furthermore, b 1 = b

The Negative Exponent Rule If b is any real number other than 0 and n is a natural number, then

The Zero Exponent Rule If b is any real number other than 0, b 0 = 1.

The Product Rule b m · b n = b m+n When multiplying exponential expressions with the same base, add the exponents. Use this sum as the exponent of the common base.

The Power Rule (Powers to Powers) (b m ) n = b mn When an exponential expression is raised to a power, multiply the exponents. Place the product of the exponents on the base and remove the parentheses.

The Quotient Rule When dividing exponential expressions with the same nonzero base, subtract the exponent in the denominator from the exponent in the numerator. Use this difference as the exponent of the common base.

Example Find the quotient of 4 3 /4 2 Solution:

Products to Powers (ab) n = a n b n When a product is raised to a power, raise each factor to the power.

Simplify: (-2y) 4. (-2y) 4 = (-2) 4 y 4 = 16y 4 Text Example Solution A.-16y 4 B.-8y 4 C.16y 4 D.8y 4

Quotients to Powers When a quotient is raised to a power, raise the numerator to that power and divide by the denominator to that power.

Example Simplify by raising the quotient (2/3) 4 to the given power. Solution:

The number 5.5 x 10 12 is written in a form called scientific notation. A number in scientific notation is expressed as a number greater than or equal to 1 and less than 10 multiplied by some power of 10. It is customary to use the multiplication symbol, x, rather than a dot in scientific notation. Scientific Notation

Text Example Write the following number in decimal notation: 2.6 X 10 7 a. 2.6 x 10 7 can be expressed in decimal notation by moving the decimal point in 2.6 seven places to the right. We need to add six zeros. 2.6 x 10 7 = 26,000,000. Solution:

Write the following number in decimal notation 1.016 X 10 -8 b. 1.016 x 10 -8 can be expressed in decimal notation by moving the decimal point in 1.016 eight places to the left. We need to add seven zeros to the right of the decimal point. 1.016 x 10 -8 = 0.00000001016.

Write each number in scientific notation. a. 4,600,000 b. 0.00023 Solution a. 4,600,000 = 4.6 x 10 ? 4.6 x 10 6 Decimal point moves 6 places b. 0.00023 = 2.3 x 10 ? 2.3 x 10 -4 Decimal point moves 4 places Text Example

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