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Exponents and Radicals. Section 1.2. 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. Exponential Notation.
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Exponents and Radicals Section 1.2
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.
Exponential Notation • an = a * a * a * a…* a (where there are n factors) • The number a is the base and n is the exponent.
Zero and Negative Exponents • If a ≠ 0 is any real number and n is a positive integer, then • a0 = 1 • a-n = 1/an
Laws of Exponents • aman = am+n • When multiplying two powers of the same base, add the exponents. • am/ an = am – n • When dividing two powers of the same base, subtract the exponents. • (am)n = amn • When raising a power to a power, multiply the exponents.
Laws of Exponents • (ab)n = anbn • When raising a product to a power, raise each factor to the power. • (a/b)n = an / bn • When raising a quotient to a power, raise both the numerator and denominator to the power. • (a/b)-n = (b/a)n • When raising a quotient to a negative power, take the reciprocal and change the power to a positive. • a-m / b-n = bm / an • To simplify a negative exponent, move it to the opposite position in the fraction. The exponent then becomes positive.
Scientific Notation • Scientific Notation—shorthand way of writing very large or very small numbers. • 4 x 1013 • 4 and 13 zero’s • 1.66 x 10-12 • 0.00000000000166
nth root • If 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.
Rational Exponents • For 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.
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.
