Algebra+and+Number+3

3. Demonstrate an understanding of powers with integral and rational exponents. [C, CN, PS, R]

3.1 Explain, using patterns, why a-n = 1an , a ≠ 0. 3.2 Explain, using patterns, why 1an  = na , n > 0. 3.3 Apply the exponent laws: <span style="background-color: transparent; color: #000000; font-family: Comic Sans MS; font-size: 16px; text-decoration: none; vertical-align: baseline;">to expressions with rational and variable bases and integral and rational exponents, and explain the reasoning.
 * <span style="background-color: transparent; color: #000000; font-family: Verdana; font-size: 13.33px; text-decoration: none; vertical-align: baseline;"><span style="background-color: transparent; color: #000000; font-family: Cambria; font-size: 16px; text-decoration: none; vertical-align: baseline;">(am <span style="background-color: transparent; color: #000000; font-family: Comic Sans MS; font-size: 16px; text-decoration: none; vertical-align: baseline;">)( <span style="background-color: transparent; color: #000000; font-family: Cambria; font-size: 16px; text-decoration: none; vertical-align: baseline;">an <span style="background-color: transparent; color: #000000; font-family: Comic Sans MS; font-size: 16px; text-decoration: none; vertical-align: baseline;">) = <span style="background-color: transparent; color: #000000; font-family: Cambria; font-size: 16px; text-decoration: none; vertical-align: baseline;">am+n
 * <span style="background-color: transparent; color: #000000; font-family: Verdana; font-size: 13.33px; text-decoration: none; vertical-align: baseline;"><span style="background-color: transparent; color: #000000; font-family: Cambria; font-size: 16px; text-decoration: none; vertical-align: baseline;">(am <span style="background-color: transparent; color: #000000; font-family: Comic Sans MS; font-size: 16px; text-decoration: none; vertical-align: baseline;">) <span style="background-color: transparent; color: #000000; font-family: Cambria; font-size: 16px; text-decoration: none; vertical-align: baseline;">÷ <span style="background-color: transparent; color: #000000; font-family: Comic Sans MS; font-size: 16px; text-decoration: none; vertical-align: baseline;">( <span style="background-color: transparent; color: #000000; font-family: Cambria; font-size: 16px; text-decoration: none; vertical-align: baseline;">an <span style="background-color: transparent; color: #000000; font-family: Comic Sans MS; font-size: 16px; text-decoration: none; vertical-align: baseline;">) = <span style="background-color: transparent; color: #000000; font-family: Cambria; font-size: 16px; text-decoration: none; vertical-align: baseline;">am-n <span style="background-color: transparent; color: #000000; font-family: Comic Sans MS; font-size: 16px; text-decoration: none; vertical-align: baseline;"> a <span style="background-color: transparent; color: #000000; font-family: Cambria; font-size: 16px; text-decoration: none; vertical-align: baseline;">≠ <span style="background-color: transparent; color: #000000; font-family: Comic Sans MS; font-size: 16px; text-decoration: none; vertical-align: baseline;">0
 * <span style="background-color: transparent; color: #000000; font-family: Verdana; font-size: 13.33px; text-decoration: none; vertical-align: baseline;"><span style="background-color: transparent; color: #000000; font-family: Cambria; font-size: 16px; text-decoration: none; vertical-align: baseline;">(a)(m)n <span style="background-color: transparent; color: #000000; font-family: Comic Sans MS; font-size: 16px; text-decoration: none; vertical-align: baseline;">= <span style="background-color: transparent; color: #000000; font-family: Cambria; font-size: 16px; text-decoration: none; vertical-align: baseline;">amn
 * <span style="background-color: transparent; color: #000000; font-family: Verdana; font-size: 13.33px; text-decoration: none; vertical-align: baseline;"><span style="background-color: transparent; color: #000000; font-family: Cambria; font-size: 16px; text-decoration: none; vertical-align: baseline;">(ab)m <span style="background-color: transparent; color: #000000; font-family: Comic Sans MS; font-size: 16px; text-decoration: none; vertical-align: baseline;">= <span style="background-color: transparent; color: #000000; font-family: Cambria; font-size: 16px; text-decoration: none; vertical-align: baseline;">am bm
 * <span style="background-color: transparent; color: #000000; font-family: Verdana; font-size: 13.33px; text-decoration: none; vertical-align: baseline;"><span style="background-color: transparent; color: #000000; font-family: Comic Sans MS; font-size: 16px; text-decoration: none; vertical-align: baseline;">( <span style="background-color: transparent; color: #000000; font-family: Cambria; font-size: 16px; text-decoration: none; vertical-align: baseline;">ab <span style="background-color: transparent; color: #000000; font-family: Comic Sans MS; font-size: 16px; text-decoration: none; vertical-align: baseline;">) <span style="background-color: transparent; color: #000000; font-family: Comic Sans MS; font-size: 9.6px; text-decoration: none; vertical-align: super;">n <span style="background-color: transparent; color: #000000; font-family: Comic Sans MS; font-size: 16px; text-decoration: none; vertical-align: baseline;"> = <span style="background-color: transparent; color: #000000; font-family: Cambria; font-size: 16px; text-decoration: none; vertical-align: baseline;">anbn <span style="background-color: transparent; color: #000000; font-family: Comic Sans MS; font-size: 16px; text-decoration: none; vertical-align: baseline;">, b <span style="background-color: transparent; color: #000000; font-family: Cambria; font-size: 16px; text-decoration: none; vertical-align: baseline;">≠0

<span style="background-color: transparent; color: #000000; font-family: Comic Sans MS; font-size: 16px; text-decoration: none; vertical-align: baseline;">3.4 Express powers with rational exponents as radicals and vice versa. <span style="background-color: transparent; color: #000000; font-family: Comic Sans MS; font-size: 16px; text-decoration: none; vertical-align: baseline;">3.5 Solve a problem that involves exponent laws or radicals. <span style="background-color: transparent; color: #000000; font-family: Comic Sans MS; font-size: 16px; text-decoration: none; vertical-align: baseline;">3.6 Identify and correct errors in a simplification of an expression that involves powers.