Chapter 6

Evaluate \(3 a b-2 b c+a b c\) when \(a=1, b=3\) and \(c=5\)

Find the value of \(4 p^{2} q r^{3}\), given that \(p=2\), \(q=\frac{1}{2}\) and \(r=1 \frac{1}{2}\)

Find the sum of \(3 x, 2 x,-x\) and \(-7 x\)

Find the sum of \(4 a, 3 b, c,-2 a,-5 b\) and \(6 c\)

Find the sum of \(5 a-2 b, 2 a+c, 4 b-5 d\) and \(b-a+3 d-4 c\)

Subtract \(2 x+3 y-4 z\) from \(x-2 y+5 z\)

Multiply \(2 a+3 b\) by \(a+b\)

Multiply \(3 x-2 y^{2}+4 x y\) by \(2 x-5 y\)

Simplify \(2 p \div 8 p q\)

Divide \(2 x^{2}+x-3\) by \(x-1\)

Simplify \(\frac{x^{3}+y^{3}}{x+y}\)

Divide \(4 a^{3}-6 a^{2} b+5 b^{3}\) by \(2 a-b\)

Simplify \(a^{3} b^{2} c \times a b^{3} c^{5}\)

Simplify \(a^{1 / 2} b^{2} c^{-2} \times a^{1 / 6} b^{1 / 2} c\)

Simplify \(\frac{a^{3} b^{2} c^{4}}{a b c^{-2}}\) and evaluate when \(a=3\), \(b=\frac{1}{8}\) and \(c=2\)

Simplify \(\frac{p^{1 / 2} q^{2} r^{2 / 3}}{p^{1 / 4} q^{1 / 2} r^{1 / 6}}\) and evaluate when \(p=16, q=9\) and \(r=4\), taking positive roots only.

Simplify \(\frac{x^{2} y^{3}+x y^{2}}{x y}\)

Simplify \(\frac{x^{2} y}{x y^{2}-x y}\)

Simplify \(\frac{a^{2} b}{a b^{2}-a^{1 / 2} b^{3}}\)

Simplify \(\left(p^{3}\right)^{1 / 2}\left(q^{2}\right)^{4}\)

Simplify \(\frac{\left(m n^{2}\right)^{3}}{\left(m^{1 / 2} n^{1 / 4}\right)^{4}}\)

Simplify \(\left(a^{3} \sqrt{b} \sqrt{c^{5}}\right)\left(\sqrt{a} \sqrt[3]{b^{2}} c^{3}\right)\) and evaluate when \(a=\frac{1}{4}, b=6\) and \(c=1\).

Simplify \(\left(a^{3} b\right)\left(a^{-4} b^{-2}\right)\), expressing the answer with positive indices only.

Simplify \(\frac{d^{2} e^{2} f^{1 / 2}}{\left(d^{3 / 2} e f^{5 / 2}\right)^{2}}\) expressing the answer with positive indices only.

Simplify \(\frac{\left(x^{2} y^{1 / 2}\right)\left(\sqrt{x} \sqrt[3]{y^{2}}\right)}{\left(x^{5} y^{3}\right)^{1 / 2}}\)

Remove the brackets and simplify the expression \((3 a+b)+2(b+c)-4(c+d)\)

Simplify \(a^{2}-(2 a-a b)-a(3 b+a)\)

Simplify \((a+b)(a-b)\)

Remove the brackets from the expression \((x-2 y)\left(3 x+y^{2}\right)\)

Simplify \((2 x-3 y)^{2}\)

Remove the brackets from the expression $$ 2\left[p^{2}-3(q+r)+q^{2}\right] $$

Remove the brackets and simplify the expression: \(2 a-[3\\{2(4 a-b)-5(a+2 b)\\}+4 a]\)

Simplify \(x(2 x-4 y)-2 x(4 x+y)\)

Factorize (a) \(x y-3 x z\) (b) \(4 a^{2}+16 a b^{3}\) (c) \(3 a^{2} b-6 a b^{2}+15 a b\)

Factorize \(a x-a y+b x-b y\)

Factorize \(2 a x-3 a y+2 b x-3 b y\)

Factorize \(x^{3}+3 x^{2}-x-3\)

Simplify \(2 a+5 a \times 3 a-a\)

Simplify \((a+5 a) \times 2 a-3 a\)

Simplify \(a+5 a \times(2 a-3 a)\)

Simplify \(a \div(5 a+2 a)-3 a\)

Simplify \(3 c+2 c \times 4 c+c \div 5 c-8 c\)

Simplify \((3 c+2 c) 4 c+c \div 5 c-8 c\)

Simplify \(3 c+2 c \times 4 c+c \div(5 c-8 c)\)

Simplify \((3 c+2 c)(4 c+c) \div(5 c-8 c)\)

Simplify \(\frac{1}{3}\) of \(3 p+4 p(3 p-p)\)

If \(y\) is directly proportional to \(x\) and \(y=2.48\) when \(x=0.4\), determine (a) the coefficient of proportionality and (b) the value of \(y\) when \(x=0.65\)

Hooke's law states that stress \(\sigma\) is directly proportional to strain \(\varepsilon\) within the elastic limit of a material. When, for mild steel, the stress is \(25 \times 10^{6}\) pascals, the strain is \(0.000125\). Determine (a) the coefficient of proportionality and (b) the value of strain when the stress is \(18 \times 10^{6}\) pascals.

The electrical resistance \(R\) of a piece of wire is inversely proportional to the cross-sectional area \(A\). When \(A=5 \mathrm{~mm}^{2}, R=7.02\) ohms. Determine (a) the coefficient of proportionality and (b) the cross-sectional area when the resistance is \(4 \mathrm{ohms}\).

Boyle's law states that at constant temperature, the volume \(V\) of a fixed mass of gas is inversely proportional to its absolute pressure \(p .\) If a gas occupies a volume of \(0.08 \mathrm{~m}^{3}\) at a pressure of \(1.5 \times 10^{6}\) pascals determine (a) the coefficient of proportionality and (b) the volume if the pressure is changed to \(4 \times 10^{6}\) pascals..