Math Magic: Simplification Exercises

Exercise 1: Simplify

(i) $\frac{4x^{2}y}{2z^{2}} \times \frac{6xz^{3}}{20y^{4}}$

Step 1: Multiply numerators and denominators:

\frac{4x^{2}y \times 6xz^{3}}{2z^{2} \times 20y^{4}} = \frac{24x^{3}yz^{3}}{40z^{2}y^{4}}

Step 2: Simplify coefficients:

\frac{24}{40} = \frac{3}{5}

Step 3: Simplify x terms:

x^{3} \text{ remains } x^{3}

Step 4: Simplify y terms:

\frac{y}{y^{4}} = y^{1-4} = y^{-3} = \frac{1}{y^{3}}

Step 5: Simplify z terms:

\frac{z^{3}}{z^{2}} = z^{3-2} = z^{1} = z

Final Answer:

\frac{3x^{3}z}{5y^{3}}

(ii) $\frac{p^{2} - 10p + 21}{p - 7} \times \frac{p^{2} + p - 12}{(p - 3)^{2}}$

Step 1: Factor all quadratics:

p^{2} - 10p + 21 = (p-7)(p-3)

p^{2} + p - 12 = (p+4)(p-3)

Step 2: Rewrite expression with factors:

\frac{(p-7)(p-3)}{p-7} \times \frac{(p+4)(p-3)}{(p-3)^{2}}

Step 3: Cancel common terms:

\frac{\cancel{(p-7)}(p-3)}{\cancel{p-7}} \times \frac{(p+4)\cancel{(p-3)}}{\cancel{(p-3)}(p-3)}

Step 4: Simplify remaining terms:

(p-3) \times \frac{p+4}{p-3}

Step 5: Final cancellation:

\frac{\cancel{(p-3)}(p+4)}{\cancel{p-3}}

Final Answer:

\[ p + 4 \]

(iii) $\frac{5t^{3}}{4t - 8} \times \frac{6t - 12}{10t}$

Step 1: Factor denominators:

\[ 4t - 8 = 4(t - 2) \]

\[ 6t - 12 = 6(t - 2) \]

Step 2: Rewrite expression:

\frac{5t^{3}}{4(t-2)} \times \frac{6(t-2)}{10t}

Step 3: Cancel (t-2) terms:

\frac{5t^{3}}{4\cancel{(t-2)}} \times \frac{6\cancel{(t-2)}}{10t}

Step 4: Multiply numerators and denominators:

\frac{5t^{3} \times 6}{4 \times 10t} = \frac{30t^{3}}{40t}

Step 5: Simplify coefficients:

\frac{30}{40} = \frac{3}{4}

Step 6: Simplify t terms:

\frac{t^{3}}{t} = t^{3-1} = t^{2}

Final Answer:

\frac{3t^{2}}{4}

Exercise 2: Simplify

(i) $\frac{x + 4}{3x + 4y} \times \frac{9x^{2} - 16y^{2}}{2x^{2} + 3x - 20}$

Step 1: Recognize difference of squares:

9x^{2} - 16y^{2} = (3x)^{2} - (4y)^{2} = (3x + 4y)(3x - 4y)

Step 2: Factor the quadratic:

2x^{2} + 3x - 20 = (2x - 5)(x + 4)

Step 3: Rewrite expression:

\frac{x + 4}{3x + 4y} \times \frac{(3x + 4y)(3x - 4y)}{(2x - 5)(x + 4)}

Step 4: Cancel common terms:

\frac{\cancel{x + 4}}{\cancel{3x + 4y}} \times \frac{\cancel{(3x + 4y)}(3x - 4y)}{(2x - 5)\cancel{(x + 4)}}

Final Answer:

\frac{3x - 4y}{2x - 5}

(ii) $\frac{x^{3} - y^{3}}{3x^{2} + 9xy + 6y^{2}} \times \frac{x^{2} + 2xy + y^{2}}{x^{2} - y^{2}}$

Step 1: Factor difference of cubes:

x^{3} - y^{3} = (x - y)(x^{2} + xy + y^{2})

Step 2: Factor denominator:

3x^{2} + 9xy + 6y^{2} = 3(x^{2} + 3xy + 2y^{2}) = 3(x + y)(x + 2y)

Step 3: Factor perfect square:

x^{2} + 2xy + y^{2} = (x + y)^{2}

Step 4: Factor difference of squares:

x^{2} - y^{2} = (x + y)(x - y)

Step 5: Rewrite expression:

\frac{(x - y)(x^{2} + xy + y^{2})}{3(x + y)(x + 2y)} \times \frac{(x + y)^{2}}{(x + y)(x - y)}

Step 6: Cancel common terms:

\frac{\cancel{(x - y)}(x^{2} + xy + y^{2})}{3\cancel{(x + y)}(x + 2y)} \times \frac{(x + y)^{\cancel{2}}}{\cancel{(x + y)}\cancel{(x - y)}}

Step 7: Simplify remaining terms:

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

Final Answer:

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

Exercise 3: Division Problems

(i) $\frac{2a^{2} + 5a + 3}{2a^{2} + 7a + 6} \div \frac{a^{2} + 6a + 5}{-5a^{2} - 35a - 50}$

Step 1: Convert division to multiplication by reciprocal:

\frac{2a^{2} + 5a + 3}{2a^{2} + 7a + 6} \times \frac{-5a^{2} - 35a - 50}{a^{2} + 6a + 5}

Step 2: Factor all quadratics:

2a^{2} + 5a + 3 = (2a + 3)(a + 1)

2a^{2} + 7a + 6 = (2a + 3)(a + 2)

a^{2} + 6a + 5 = (a + 5)(a + 1)

-5a^{2} - 35a - 50 = -5(a^{2} + 7a + 10) = -5(a + 5)(a + 2)

Step 3: Rewrite expression:

\frac{(2a + 3)(a + 1)}{(2a + 3)(a + 2)} \times \frac{-5(a + 5)(a + 2)}{(a + 5)(a + 1)}

Step 4: Cancel common terms:

\frac{\cancel{(2a + 3)}\cancel{(a + 1)}}{\cancel{(2a + 3)}\cancel{(a + 2)}} \times \frac{-5\cancel{(a + 5)}\cancel{(a + 2)}}{\cancel{(a + 5)}\cancel{(a + 1)}}

Final Answer:

\[ -5 \]

(ii) $\frac{b^{2} + 3b - 28}{b^{2} + 4b + 4} \div \frac{b^{2} - 49}{b^{2} - 5b - 14}$

Step 1: Convert division to multiplication by reciprocal:

\frac{b^{2} + 3b - 28}{b^{2} + 4b + 4} \times \frac{b^{2} - 5b - 14}{b^{2} - 49}

Step 2: Factor all quadratics:

b^{2} + 3b - 28 = (b + 7)(b - 4)

b^{2} + 4b + 4 = (b + 2)^{2}

b^{2} - 49 = (b + 7)(b - 7)

b^{2} - 5b - 14 = (b - 7)(b + 2)

Step 3: Rewrite expression:

\frac{(b + 7)(b - 4)}{(b + 2)^{2}} \times \frac{(b - 7)(b + 2)}{(b + 7)(b - 7)}

Step 4: Cancel common terms:

\frac{\cancel{(b + 7)}(b - 4)}{(b + 2)^{\cancel{2}}} \times \frac{\cancel{(b - 7)}\cancel{(b + 2)}}{\cancel{(b + 7)}\cancel{(b - 7)}}

Step 5: Simplify remaining terms:

\frac{b - 4}{b + 2}

Final Answer:

\frac{b - 4}{b + 2}

(iii) $\frac{x + 2}{4y} \div \frac{x^{2} - x - 6}{12y^{2}}$

Step 1: Convert division to multiplication by reciprocal:

\frac{x + 2}{4y} \times \frac{12y^{2}}{x^{2} - x - 6}

Step 2: Factor quadratic:

x^{2} - x - 6 = (x - 3)(x + 2)

Step 3: Rewrite expression:

\frac{x + 2}{4y} \times \frac{12y^{2}}{(x - 3)(x + 2)}

Step 4: Cancel common terms:

\frac{\cancel{x + 2}}{4y} \times \frac{12y^{2}}{(x - 3)\cancel{(x + 2)}}

Step 5: Multiply numerators and denominators:

\frac{12y^{2}}{4y(x - 3)}

Step 6: Simplify coefficients:

\frac{12}{4} = 3

Step 7: Simplify y terms:

\frac{y^{2}}{y} = y^{2-1} = y

Final Answer:

\frac{3y}{x - 3}

(iv) $\frac{12t^{2} - 22t + 8}{3t} \div \frac{3t^{2} + 2t - 8}{2t^{2} + 4t}$

Step 1: Convert division to multiplication by reciprocal:

\frac{12t^{2} - 22t + 8}{3t} \times \frac{2t^{2} + 4t}{3t^{2} + 2t - 8}

Step 2: Factor all quadratics:

12t^{2} - 22t + 8 = 2(6t^{2} - 11t + 4) = 2(3t - 4)(2t - 1)

2t^{2} + 4t = 2t(t + 2)

3t^{2} + 2t - 8 = (3t - 4)(t + 2)

Step 3: Rewrite expression:

\frac{2(3t - 4)(2t - 1)}{3t} \times \frac{2t(t + 2)}{(3t - 4)(t + 2)}

Step 4: Cancel common terms:

\frac{2\cancel{(3t - 4)}(2t - 1)}{3t} \times \frac{2t\cancel{(t + 2)}}{\cancel{(3t - 4)}\cancel{(t + 2)}}

Step 5: Multiply numerators and denominators:

\frac{2(2t - 1) \cdot 2t}{3t} = \frac{4t(2t - 1)}{3t}

Step 6: Cancel t terms:

\frac{4\cancel{t}(2t - 1)}{3\cancel{t}}

Final Answer:

\frac{4(2t - 1)}{3}

Exercise 4: Advanced Problem

If $x = \frac{a^{2} + 3a - 4}{3a^{2} - 3}$ and $y = \frac{a^{2} + 2a - 8}{2a^{2} - 2a - 4}$ find the value of $x^{2}y^{-2}$

Step 1: Simplify x:

x = \frac{a^{2} + 3a - 4}{3a^{2} - 3} = \frac{(a + 4)(a - 1)}{3(a^{2} - 1)} = \frac{(a + 4)(a - 1)}{3(a + 1)(a - 1)}

x = \frac{a + 4}{3(a + 1)} \quad \text{(after canceling } (a-1)\text{)}

Step 2: Simplify y:

y = \frac{a^{2} + 2a - 8}{2a^{2} - 2a - 4} = \frac{(a + 4)(a - 2)}{2(a^{2} - a - 2)} = \frac{(a + 4)(a - 2)}{2(a - 2)(a + 1)}

y = \frac{a + 4}{2(a + 1)} \quad \text{(after canceling } (a-2)\text{)}

Step 3: Find y⁻²:

y^{-2} = \left( \frac{a + 4}{2(a + 1)} \right)^{-2} = \left( \frac{2(a + 1)}{a + 4} \right)^{2}

Step 4: Find x²:

x^{2} = \left( \frac{a + 4}{3(a + 1)} \right)^{2}

Step 5: Multiply x² and y⁻²:

x^{2}y^{-2} = \left( \frac{a + 4}{3(a + 1)} \right)^{2} \times \left( \frac{2(a + 1)}{a + 4} \right)^{2}

Step 6: Simplify:

= \frac{(a + 4)^{2}}{9(a + 1)^{2}} \times \frac{4(a + 1)^{2}}{(a + 4)^{2}}

Step 7: Cancel terms:

= \frac{\cancel{(a + 4)^{2}}}{9\cancel{(a + 1)^{2}}} \times \frac{4\cancel{(a + 1)^{2}}}{\cancel{(a + 4)^{2}}} = \frac{4}{9}

Final Answer:

\frac{4}{9}

Exercise 5: Polynomial Division

If a polynomial $p(x) = x^{2} - 5x - 14$ is divided by another polynomial $\[ q(x) \]$ we get $\frac{x - 7}{x + 2}$ , find $\[ q(x) \]$ .

Step 1: Set up the division equation:

\frac{p(x)}{q(x)} = \frac{x - 7}{x + 2}

Step 2: Solve for q(x):

q(x) = p(x) \times \frac{x + 2}{x - 7}

Step 3: Factor p(x):

p(x) = x^{2} - 5x - 14 = (x - 7)(x + 2)

Step 4: Substitute factored form:

q(x) = \frac{(x - 7)(x + 2)}{1} \times \frac{x + 2}{x - 7}

Step 5: Cancel common terms:

q(x) = \frac{\cancel{(x - 7)}(x + 2)}{1} \times \frac{x + 2}{\cancel{x - 7}}

Step 6: Multiply remaining terms:

q(x) = (x + 2)(x + 2) = (x + 2)^{2}

Final Answer:

q(x) = (x + 2)^{2} \quad \text{or} \quad x^{2} + 4x + 4