This lesson brings no new theory: it is a set of exercises on everything from lessons 20-24. It is worth working through all of them, because every later topic in this course - equations, functions, derivatives - rests on these manipulations. If one type of exercise gives you trouble, go back to the lesson it comes from now rather than meeting the same gap three chapters later.
Algebra - review exercises Simplify each of the following expressions.
a) 2 a b − 4 a c + b a − 2 c b + 3 b a = 6 a b − 4 a c − 2 c b = = 2 ( 3 a b − 2 a c − c b ) \begin{aligned}\text{a)}\quad 2ab - 4ac + ba - 2cb + 3ba &= 6ab - 4ac - 2cb = \\&= 2(3ab - 2ac - cb)\end{aligned} a) 2 ab − 4 a c + ba − 2 c b + 3 ba = 6 ab − 4 a c − 2 c b = = 2 ( 3 ab − 2 a c − c b ) b) 3 x 2 y z − z x 2 y + 4 y x z 2 − 2 x 2 z y + 3 z 2 y x − 3 z y 2 = = 0 ⋅ x 2 y z + 7 z 2 y x − 3 z y 2 = 7 z 2 y x − 3 z y 2 = = y z ( 7 x z − 3 y ) \begin{aligned}\text{b)}\quad 3x^{2}yz - zx^{2}y + 4yxz^{2} - 2x^{2}zy + 3z^{2}yx - 3zy^{2} &= \\= 0\cdot x^{2}yz + 7z^{2}yx - 3zy^{2} &= 7z^{2}yx - 3zy^{2} = \\&= yz(7xz - 3y)\end{aligned} b) 3 x 2 y z − z x 2 y + 4 y x z 2 − 2 x 2 z y + 3 z 2 y x − 3 z y 2 = 0 ⋅ x 2 y z + 7 z 2 y x − 3 z y 2 = = 7 z 2 y x − 3 z y 2 = = y z ( 7 x z − 3 y ) c) c p ⋅ c − q : c − 2 = c p + ( − q ) − ( − 2 ) = c p − q + 2 \text{c)}\quad c^{p}\cdot c^{-q} : c^{-2} = c^{\,p + (-q) - (-2)} = c^{\,p - q + 2} c) c p ⋅ c − q : c − 2 = c p + ( − q ) − ( − 2 ) = c p − q + 2 d) ( x 1 2 ) − 2 3 : ( y 3 4 ) 2 ⋅ ( x 3 5 ) − 5 3 ( x 1 4 ) − 1 ⋅ ( y 1 3 ) 6 = x − 1 3 : y 3 2 ⋅ x − 1 x − 1 4 ⋅ y 2 = x − 13 12 ⋅ y − 7 2 \text{d)}\quad \frac{\left(x^{\frac{1}{2}}\right)^{-\frac{2}{3}} : \left(y^{\frac{3}{4}}\right)^{2}\cdot\left(x^{\frac{3}{5}}\right)^{-\frac{5}{3}}}{\left(x^{\frac{1}{4}}\right)^{-1}\cdot\left(y^{\frac{1}{3}}\right)^{6}} = \frac{x^{-\frac{1}{3}} : y^{\frac{3}{2}}\cdot x^{-1}}{x^{-\frac{1}{4}}\cdot y^{2}} = x^{-\frac{13}{12}}\cdot y^{-\frac{7}{2}} d) ( x 4 1 ) − 1 ⋅ ( y 3 1 ) 6 ( x 2 1 ) − 3 2 : ( y 4 3 ) 2 ⋅ ( x 5 3 ) − 3 5 = x − 4 1 ⋅ y 2 x − 3 1 : y 2 3 ⋅ x − 1 = x − 12 13 ⋅ y − 2 7 − 1 3 + ( − 1 ) − ( − 1 4 ) = − 4 3 + 1 4 = − 16 12 + 3 12 = − 13 12 -\tfrac{1}{3} + (-1) - \left(-\tfrac{1}{4}\right) = -\tfrac{4}{3} + \tfrac{1}{4} = -\tfrac{16}{12} + \tfrac{3}{12} = -\tfrac{13}{12} − 3 1 + ( − 1 ) − ( − 4 1 ) = − 3 4 + 4 1 = − 12 16 + 12 3 = − 12 13 − 3 2 − 2 = − 7 2 -\tfrac{3}{2} - 2 = -\tfrac{7}{2} − 2 3 − 2 = − 2 7 Remove the brackets from each of the following expressions.
a) 2 f ( 3 g − 4 h ) ( g + 2 h ) = 2 f ( 3 g 2 + 6 g h − 4 g h − 8 h 2 ) = = 2 f ( 3 g 2 + 2 g h − 8 h 2 ) = = 6 f g 2 + 4 f g h − 16 f h 2 \begin{aligned}\text{a)}\quad 2f(3g - 4h)(g + 2h) &= 2f(3g^{2} + 6gh - 4gh - 8h^{2}) = \\&= 2f(3g^{2} + 2gh - 8h^{2}) = \\&= 6fg^{2} + 4fgh - 16fh^{2}\end{aligned} a) 2 f ( 3 g − 4 h ) ( g + 2 h ) = 2 f ( 3 g 2 + 6 g h − 4 g h − 8 h 2 ) = = 2 f ( 3 g 2 + 2 g h − 8 h 2 ) = = 6 f g 2 + 4 f g h − 16 f h 2 b) ( 5 x − 6 y ) ( 2 x + 6 y ) ( 5 x − y ) = ( 10 x 2 + 30 x y − 12 x y − 36 y 2 ) ( 5 x − y ) = = ( 10 x 2 + 18 x y − 36 y 2 ) ( 5 x − y ) = = 50 x 3 − 10 x 2 y + 90 x 2 y − 18 x y 2 − 180 x y 2 + 36 y 3 = = 50 x 3 + 80 x 2 y − 198 x y 2 + 36 y 3 \begin{aligned}\text{b)}\quad (5x - 6y)(2x + 6y)(5x - y) &= (10x^{2} + 30xy - 12xy - 36y^{2})(5x - y) = \\&= (10x^{2} + 18xy - 36y^{2})(5x - y) = \\&= 50x^{3} - 10x^{2}y + 90x^{2}y - 18xy^{2} - 180xy^{2} + 36y^{3} = \\&= 50x^{3} + 80x^{2}y - 198xy^{2} + 36y^{3}\end{aligned} b) ( 5 x − 6 y ) ( 2 x + 6 y ) ( 5 x − y ) = ( 10 x 2 + 30 x y − 12 x y − 36 y 2 ) ( 5 x − y ) = = ( 10 x 2 + 18 x y − 36 y 2 ) ( 5 x − y ) = = 50 x 3 − 10 x 2 y + 90 x 2 y − 18 x y 2 − 180 x y 2 + 36 y 3 = = 50 x 3 + 80 x 2 y − 198 x y 2 + 36 y 3 c) 4 { 3 [ p − 2 ( q − 3 ) ] − 3 ( r − 2 ) } = 4 { 3 [ p − 2 q + 6 ] − 3 r + 6 } = = 4 { 3 p − 6 q + 18 − 3 r + 6 } = = 4 ( 3 p − 6 q + 24 − 3 r ) = = 12 p − 24 q − 12 r + 96 \begin{aligned}\text{c)}\quad 4\{3[p - 2(q - 3)] - 3(r - 2)\} &= 4\{3[p - 2q + 6] - 3r + 6\} = \\&= 4\{3p - 6q + 18 - 3r + 6\} = \\&= 4(3p - 6q + 24 - 3r) = \\&= 12p - 24q - 12r + 96\end{aligned} c) 4 { 3 [ p − 2 ( q − 3 )] − 3 ( r − 2 )} = 4 { 3 [ p − 2 q + 6 ] − 3 r + 6 } = = 4 { 3 p − 6 q + 18 − 3 r + 6 } = = 4 ( 3 p − 6 q + 24 − 3 r ) = = 12 p − 24 q − 12 r + 96 Evaluate using a calculator or by applying the change-of-base formula where necessary.
a) log 0.0270 ≈ − 1.569 \text{a)}\quad \log 0.0270 \approx -1.569 a) log 0.0270 ≈ − 1.569 b) ln 47.89 ≈ 3.869 \text{b)}\quad \ln 47.89 \approx 3.869 b) ln 47.89 ≈ 3.869 c) log 7 126.4 = log 7 10 ⋅ log 10 126.4 = 1 log 10 7 ⋅ log 10 126.4 ≈ 2.102 0.845 ≈ 2.487 \text{c)}\quad \log_{7}126.4 = \log_{7}10\cdot\log_{10}126.4 = \frac{1}{\log_{10}7}\cdot\log_{10}126.4 \approx \frac{2.102}{0.845} \approx 2.487 c) log 7 126.4 = log 7 10 ⋅ log 10 126.4 = log 10 7 1 ⋅ log 10 126.4 ≈ 0.845 2.102 ≈ 2.487 Write the following expressions in logarithmic form.
a) V = π h 4 ( D − h ) ( D + h ) log V = log π + log h − log 4 + log ( D − h ) + log ( D + h ) \begin{aligned}\text{a)}\quad V &= \frac{\pi h}{4}(D - h)(D + h) \\\log V &= \log \pi + \log h - \log 4 + \log(D - h) + \log(D + h)\end{aligned} a) V log V = 4 π h ( D − h ) ( D + h ) = log π + log h − log 4 + log ( D − h ) + log ( D + h ) b) P = 1 16 ( 2 d − 1 ) 2 N S = 2 − 4 ( 2 d − 1 ) 2 N S log P = log 2 − 4 + log ( 2 d − 1 ) 2 + log N + log S log P = − 4 log 2 + 2 log ( 2 d − 1 ) + log N + 1 2 log S \begin{aligned}\text{b)}\quad P &= \tfrac{1}{16}(2d - 1)^{2}N\sqrt{S} = 2^{-4}(2d - 1)^{2}N\sqrt{S} \\\log P &= \log 2^{-4} + \log(2d - 1)^{2} + \log N + \log\sqrt{S} \\\log P &= -4\log 2 + 2\log(2d - 1) + \log N + \tfrac{1}{2}\log S\end{aligned} b) P log P log P = 16 1 ( 2 d − 1 ) 2 N S = 2 − 4 ( 2 d − 1 ) 2 N S = log 2 − 4 + log ( 2 d − 1 ) 2 + log N + log S = − 4 log 2 + 2 log ( 2 d − 1 ) + log N + 2 1 log S Rewrite the following expressions without using logarithms.
a) log x = log P + 2 log Q − log k − 3 3 = log P + log Q 2 − log k − log x = log P Q 2 x k P Q 2 x k = 10 3 ⇒ P Q 2 = 1000 x k ⇒ x = P Q 2 1000 k \begin{aligned}\text{a)}\quad \log x &= \log P + 2\log Q - \log k - 3 \\3 &= \log P + \log Q^{2} - \log k - \log x = \log\frac{PQ^{2}}{xk} \\\frac{PQ^{2}}{xk} &= 10^{3} \quad\Rightarrow\quad PQ^{2} = 1000xk \quad\Rightarrow\quad x = \frac{PQ^{2}}{1000k}\end{aligned} a) log x 3 x k P Q 2 = log P + 2 log Q − log k − 3 = log P + log Q 2 − log k − log x = log x k P Q 2 = 1 0 3 ⇒ P Q 2 = 1000 x k ⇒ x = 1000 k P Q 2 b) log R = 1 + 1 3 log M + 3 log S 1 = log R − log M 3 − log S 3 = log R S 3 M 3 R S 3 M 3 = 10 ⇒ R = 10 S 3 M 3 \begin{aligned}\text{b)}\quad \log R &= 1 + \tfrac{1}{3}\log M + 3\log S \\1 &= \log R - \log\sqrt[3]{M} - \log S^{3} = \log\frac{R}{S^{3}\sqrt[3]{M}} \\\frac{R}{S^{3}\sqrt[3]{M}} &= 10 \quad\Rightarrow\quad R = 10S^{3}\sqrt[3]{M}\end{aligned} b) log R 1 S 3 3 M R = 1 + 3 1 log M + 3 log S = log R − log 3 M − log S 3 = log S 3 3 M R = 10 ⇒ R = 10 S 3 3 M c) ln P = 1 2 ln ( Q + 1 ) − 3 ln R + 2 2 = ln P − ln Q + 1 + ln R 3 = ln P R 3 Q + 1 P R 3 Q + 1 = e 2 ⇒ P R 3 = e 2 Q + 1 ⇒ P = e 2 Q + 1 R 3 \begin{aligned}\text{c)}\quad \ln P &= \tfrac{1}{2}\ln(Q + 1) - 3\ln R + 2 \\2 &= \ln P - \ln\sqrt{Q + 1} + \ln R^{3} = \ln\frac{PR^{3}}{\sqrt{Q + 1}} \\\frac{PR^{3}}{\sqrt{Q + 1}} &= e^{2} \quad\Rightarrow\quad PR^{3} = e^{2}\sqrt{Q + 1} \quad\Rightarrow\quad P = \frac{e^{2}\sqrt{Q + 1}}{R^{3}}\end{aligned} c) ln P 2 Q + 1 P R 3 = 2 1 ln ( Q + 1 ) − 3 ln R + 2 = ln P − ln Q + 1 + ln R 3 = ln Q + 1 P R 3 = e 2 ⇒ P R 3 = e 2 Q + 1 ⇒ P = R 3 e 2 Q + 1 Multiply and simplify the resulting expressions.
a) ( 3 x − 1 ) ( x 2 − x − 1 ) = 3 x 3 − 3 x 2 − 3 x − x 2 + x + 1 = = 3 x 3 − 4 x 2 − 2 x + 1 \begin{aligned}\text{a)}\quad (3x - 1)(x^{2} - x - 1) &= 3x^{3} - 3x^{2} - 3x - x^{2} + x + 1 = \\&= 3x^{3} - 4x^{2} - 2x + 1\end{aligned} a) ( 3 x − 1 ) ( x 2 − x − 1 ) = 3 x 3 − 3 x 2 − 3 x − x 2 + x + 1 = = 3 x 3 − 4 x 2 − 2 x + 1 b) ( a 2 + 2 a + 2 ) ( 3 a 2 + 4 a + 4 ) = 3 a 4 + 4 a 3 + 4 a 2 + 6 a 3 + 8 a 2 + 8 a + 6 a 2 + 8 a + 8 = = 3 a 4 + 10 a 3 + 18 a 2 + 16 a + 8 \begin{aligned}\text{b)}\quad (a^{2} + 2a + 2)(3a^{2} + 4a + 4) &= 3a^{4} + 4a^{3} + 4a^{2} + 6a^{3} + 8a^{2} + 8a + 6a^{2} + 8a + 8 = \\&= 3a^{4} + 10a^{3} + 18a^{2} + 16a + 8\end{aligned} b) ( a 2 + 2 a + 2 ) ( 3 a 2 + 4 a + 4 ) = 3 a 4 + 4 a 3 + 4 a 2 + 6 a 3 + 8 a 2 + 8 a + 6 a 2 + 8 a + 8 = = 3 a 4 + 10 a 3 + 18 a 2 + 16 a + 8 Simplify each of the following expressions.
a) x 3 y 2 : x y 3 = x 3 y 2 ⋅ y 3 x = x 3 y 3 x y 2 = x 2 y \text{a)}\quad \frac{x^{3}}{y^{2}} : \frac{x}{y^{3}} = \frac{x^{3}}{y^{2}}\cdot\frac{y^{3}}{x} = \frac{x^{3}y^{3}}{xy^{2}} = x^{2}y a) y 2 x 3 : y 3 x = y 2 x 3 ⋅ x y 3 = x y 2 x 3 y 3 = x 2 y b) a b c : a c b ⋅ b c a = a b c ⋅ b a c ⋅ b c a = a b 3 c a 2 c 2 = b 3 a c \text{b)}\quad \frac{ab}{c} : \frac{ac}{b}\cdot\frac{bc}{a} = \frac{ab}{c}\cdot\frac{b}{ac}\cdot\frac{bc}{a} = \frac{ab^{3}c}{a^{2}c^{2}} = \frac{b^{3}}{ac} b) c ab : b a c ⋅ a b c = c ab ⋅ a c b ⋅ a b c = a 2 c 2 a b 3 c = a c b 3 Perform the division.
a)
( x 2 + 2 x − 3 ) : ( x − 1 ) = x + 3 (x^{2} + 2x - 3) : (x - 1) = x + 3 ( x 2 + 2 x − 3 ) : ( x − 1 ) = x + 3 x 2 x^{2} x 2 − x - x − x 0 3 x 3x 3 x − 3 - 3 − 3 3 x 3x 3 x − 3 - 3 − 3 0 0 0
( x − 1 ) ( x + 3 ) = x 2 + 2 x − 3 (x - 1)(x + 3) = x^{2} + 2x - 3 ( x − 1 ) ( x + 3 ) = x 2 + 2 x − 3 b)
( n 3 + 0 n 2 + 0 n + 27 ) : ( n + 3 ) = n 2 − 3 n + 9 (n^{3} + 0n^{2} + 0n + 27) : (n + 3) = n^{2} - 3n + 9 ( n 3 + 0 n 2 + 0 n + 27 ) : ( n + 3 ) = n 2 − 3 n + 9 n 3 n^{3} n 3 + 3 n 2 + 3n^{2} + 3 n 2 0 − 3 n 2 - 3n^{2} − 3 n 2 + 0 n + 0n + 0 n − 3 n 2 - 3n^{2} − 3 n 2 − 9 n - 9n − 9 n 0 9 n 9n 9 n + 27 + 27 + 27 9 n 9n 9 n + 27 + 27 + 27 0 0 0
( n + 3 ) ( n 2 − 3 n + 9 ) = n 3 + 0 n 2 + 0 n + 27 (n + 3)(n^{2} - 3n + 9) = n^{3} + 0n^{2} + 0n + 27 ( n + 3 ) ( n 2 − 3 n + 9 ) = n 3 + 0 n 2 + 0 n + 27 c)
( 3 a 3 + 2 a 2 + 0 a + 1 ) : ( a + 1 ) = 3 a 2 − a + 1 (3a^{3} + 2a^{2} + 0a + 1) : (a + 1) = 3a^{2} - a + 1 ( 3 a 3 + 2 a 2 + 0 a + 1 ) : ( a + 1 ) = 3 a 2 − a + 1 3 a 3 3a^{3} 3 a 3 + 3 a 2 + 3a^{2} + 3 a 2 0 − a 2 - a^{2} − a 2 + 0 a + 0a + 0 a − a 2 - a^{2} − a 2 − a - a − a 0 a a a + 1 + 1 + 1 a a a + 1 + 1 + 1 0 0 0
( a + 1 ) ( 3 a 2 − a + 1 ) = 3 a 3 + 2 a 2 + 0 a + 1 (a + 1)(3a^{2} - a + 1) = 3a^{3} + 2a^{2} + 0a + 1 ( a + 1 ) ( 3 a 2 − a + 1 ) = 3 a 3 + 2 a 2 + 0 a + 1 Factor the following expressions.
a) 36 x 3 y 2 − 8 x 2 y = 4 x 2 y ( 9 x y − 2 ) \text{a)}\quad 36x^{3}y^{2} - 8x^{2}y = 4x^{2}y(9xy - 2) a) 36 x 3 y 2 − 8 x 2 y = 4 x 2 y ( 9 x y − 2 ) b) x 3 + 3 x 2 y + 2 x y 2 + 6 y 3 = x 2 ( x + 3 y ) + 2 y 2 ( x + 3 y ) = = ( x + 3 y ) ( x 2 + 2 y 2 ) \begin{aligned}\text{b)}\quad x^{3} + 3x^{2}y + 2xy^{2} + 6y^{3} &= x^{2}(x + 3y) + 2y^{2}(x + 3y) = \\&= (x + 3y)(x^{2} + 2y^{2})\end{aligned} b) x 3 + 3 x 2 y + 2 x y 2 + 6 y 3 = x 2 ( x + 3 y ) + 2 y 2 ( x + 3 y ) = = ( x + 3 y ) ( x 2 + 2 y 2 ) c) 4 x 2 + 12 x + 9 = ( 2 x + 3 ) 2 \text{c)}\quad 4x^{2} + 12x + 9 = (2x + 3)^{2} c) 4 x 2 + 12 x + 9 = ( 2 x + 3 ) 2 d) ( 3 x + 4 y ) 2 − ( 2 x − y ) 2 = [ 3 x + 4 y − ( 2 x − y ) ] [ 3 x + 4 y + ( 2 x − y ) ] = = ( 3 x + 4 y − 2 x + y ) ( 3 x + 4 y + 2 x − y ) = = ( x + 5 y ) ( 5 x + 3 y ) \begin{aligned}\text{d)}\quad (3x + 4y)^{2} - (2x - y)^{2} &= [3x + 4y - (2x - y)][3x + 4y + (2x - y)] = \\&= (3x + 4y - 2x + y)(3x + 4y + 2x - y) = \\&= (x + 5y)(5x + 3y)\end{aligned} d) ( 3 x + 4 y ) 2 − ( 2 x − y ) 2 = [ 3 x + 4 y − ( 2 x − y )] [ 3 x + 4 y + ( 2 x − y )] = = ( 3 x + 4 y − 2 x + y ) ( 3 x + 4 y + 2 x − y ) = = ( x + 5 y ) ( 5 x + 3 y ) e) x 2 + 10 x + 24 = ( x + 6 ) ( x + 4 ) \text{e)}\quad x^{2} + 10x + 24 = (x + 6)(x + 4) e) x 2 + 10 x + 24 = ( x + 6 ) ( x + 4 ) f) x 2 − 10 x + 16 = ( x − 8 ) ( x − 2 ) \text{f)}\quad x^{2} - 10x + 16 = (x - 8)(x - 2) f) x 2 − 10 x + 16 = ( x − 8 ) ( x − 2 ) g) x 2 − 5 x − 36 = ( x + 4 ) ( x − 9 ) \text{g)}\quad x^{2} - 5x - 36 = (x + 4)(x - 9) g) x 2 − 5 x − 36 = ( x + 4 ) ( x − 9 ) h) 6 x 2 + 5 x − 6 = 6 x 2 + 9 x − 4 x − 6 = 3 x ( 2 x + 3 ) − 2 ( 2 x + 3 ) = = ( 2 x + 3 ) ( 3 x − 2 ) \begin{aligned}\text{h)}\quad 6x^{2} + 5x - 6 &= 6x^{2} + 9x - 4x - 6 = 3x(2x + 3) - 2(2x + 3) = \\&= (2x + 3)(3x - 2)\end{aligned} h) 6 x 2 + 5 x − 6 = 6 x 2 + 9 x − 4 x − 6 = 3 x ( 2 x + 3 ) − 2 ( 2 x + 3 ) = = ( 2 x + 3 ) ( 3 x − 2 )