By John Edward Breen, American Association of State Highway and Transportation Officials, United States. Federal Highway Administration
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Households with Internet service for selected years were as follows. 2% For each of these years, use the equation to ﬁnd the predicted percent of households with Internet service. 8%. What percent increase does the equation predict from 2008 to 2011? Why do you think the 2008–2011 change is so different from the 2001–2004 change? (c) Why is it reasonable for a formula such as this to have an upper limit that cannot be exceeded? Use large t-values in the formula to discover this formula’s upper limit.
Only even roots of negatives are not real numbers. Thus 1Ϫ64 and 1 Ϫ1 are not real numbers. 1 1 3 5 2. (a) x1͞3 ϭ 1 x, x2͞5 ϭ 1 x2, xϪ3͞2 ϭ 3͞2 ϭ x 1x3 1 1 4 3 ϭ ϭ xϪ1͞2 (b) 1 x ϭ x3͞4, 1x x1͞2 3 3 3. 32867 4. (a) x ؒ x1͞3 ؒ x3 ϭ x1ϩ1͞3ϩ3 ϭ x13͞3 (b) x2 Ϭ x1͞2 ϭ x2Ϫ1͞2 ϭ x3͞2 (c) (xϪ2͞3)Ϫ3 ϭ x(Ϫ2͞3)(Ϫ3) ϭ x2 (d) xϪ3͞2 ؒ x1͞2 ϭ xϪ3͞2ϩ1͞2 ϭ xϪ1 x4 3͞2 (x4)3͞2 x6 (e) xϪ3͞2 ؒ x ϭ xϪ3͞2ϩ1 ϭ xϪ1͞2 (f) a 2 b ϭ 2 3͞2 ϭ 3 y (y ) y 8x2͞3 ϭ 8x2͞3 ؒ x1͞3 ϭ 8x2͞3ϩ1͞3 ϭ 8x 5. (a) False. xϪ1͞3 4 (16x8y)3͞4 ϭ 163͞4(x8)3͞4y3͞4 ϭ ( 1 16)3x6y3͞4 ϭ 8x6y3͞4 (b) False.
A) Write as radicals: x , x , x 1 1 4 ϭ ? ϭ x? , 1x x 3. Evaluate the following. 15 (a) 82͞3 (b) (Ϫ8)2͞3 (c) 8Ϫ2͞3 (d) Ϫ8Ϫ2͞3 (e) 1 71 4. Complete the following. (a) x ؒ x1͞3 ؒ x3 ϭ x? (b) x2 Ϭ x1͞2 ϭ x? (c) (xϪ2͞3)Ϫ3 ϭ x? x4 3͞2 (d) xϪ3͞2 ؒ x1͞2 ϭ x? (e) xϪ3͞2 ؒ x ϭ x? (f) a 2 b ϭ ? y 5. True or false: 8x2͞3 (a) Ϫ1͞3 ϭ 4x (b) (16x8y)3͞4 ϭ 12x6y3͞4 x y3 1͞3 y x2 Ϫ1͞3 ϭ a 2 b ϭ 2͞3 (c) a 3 b y x x We can perform operations with radicals by ﬁrst rewriting in exponential form, performing the operations with exponents, and then converting the answer back to radical form.