2 Objectives Define integer exponents and exponential notation. Define zero and negative exponents.Identify laws of exponents.Write numbers using scientific notation.Define nth roots and rational exponents.Define Rationalizing the Denominator
3 Exponential Notationan = a * a * a * a…* a (where there are n factors)The number a is the base and n is the exponent.
4 Zero and Negative Exponents If a ≠ 0 is any real number and n is a positive integer, thena0 = 1a-n = 1/an
5 Laws of Exponents aman = am+n am/ an = am – n (am)n = amn When multiplying two powers of the same base, add the exponents.am/ an = am – nWhen dividing two powers of the same base, subtract the exponents.(am)n = amnWhen raising a power to a power, multiply the exponents.
6 Laws of Exponents (ab)n = anbn (a/b)n = an / bn (a/b)-n = (b/a)n When raising a product to a power, raise each factor to the power.(a/b)n = an / bnWhen raising a quotient to a power, raise both the numerator and denominator to the power.(a/b)-n = (b/a)nWhen raising a quotient to a negative power, take the reciprocal and change the power to a positive.a-m / b-n = bm / anTo simplify a negative exponent, move it to the opposite position in the fraction. The exponent then becomes positive.
7 nth rootIf n is any positive integer, then the principal nth root of a is defined as:If n is even, a and b must be positive.
9 Rational ExponentsFor any rational exponent m/n in lowest terms, where m and n are integers and n>0, we define:If n is even, then we require that a ≥ 0.
10 7.1 – Radicals Radical Expressions The above symbol represents the positive or principal root of a number.The symbol represents the negative root of a number.
11 7.1 – Radicals Square Roots A square root of any positive number has two roots – one is positive and the other is negative.If a is a positive number, thenis the positive square root of a andis the negative square root of a.Examples:non-real #
12 Simplifying RadicalsThe radicand has no factor raised to a power greater than or equal to the index number.The radicand has no fractions.No denominator contains a radical.Exponents in the radicand and the index of the radical have no common factor.All indicated operations have been performed
13 Simplify each expression: Simplify each radical first and then combine.
14 Rationalizing the Denominator We don’t like to have radicals in the denominator, so we must rationalize to get rid of it.Rationalizing the denominator is multiplying the top and bottom of the expression by the radical you are trying to eliminate and then simplifying.
15 Rationalizing the denominator This cannot be divided which leaves the radical in the denominator. We do not leave radicals in the denominator. So we need to rationalize by multiplying the fraction by something so we can eliminate the radical in the denominator.42 cannot be simplified, so we are finished.