Learn123

Manipulating Algebraic Expressions

Collecting Terms
1) Simplify...
(i) 3x + 6x
(ii) 8x + 4y - 5x - 7
(iii) 10x - x
(iv) -9x + x
(v) 13 + 5t + 6y - t - y - 2
(vi) a - 4a
(vii) 8x - 5x + 3 + 2y - y - 1
(viii) 8x + 3x + 4x - 6x
(ix) 3p + 3 + 5p - 7 - 7p - 9
(x) 2k + 3m + 8n - 3k - 6m - 5n + 2k - m + n
(xi) 2a + 3b - 4c + 4a - 5b - 8c - 6a + 2b + 12c
(xii) r - 2s - t + 2r - 5t - 6r - 7t - s + 5s - 2t + 4r
(xiii) 7x + 3y³ - 2xy + 2xy - 5x - 6y³

Removing Brackets
2) Expand the brackets and simplify...
(i) 4(3 + x)
(ii) 2(5x + 7)
(iii) 8(3x + 2y) + 4(x + 3y)
(iv) 2(3a - 4b + 5c) - 3(2a - 5b - c)
(v) 6(2p - 3q + 4r) - 5(2p - 6p - 3r) - 3(p - 4q + 2r)
(vi) 4(l + w + h) + 3(2l - w - 2h) + 5w
(vii) 5u - 6(w - v) + 2(3u + 4w - v) - 11u
(viii) a(b + c) + a(b - c)
(ix) k(m + n) - m(k + n)
(x) p(2q + r + 3s) - pr - s(3p + q)
(xi) x(x - 2) - x(x - 6) + 8
(xii) x(x - 1) + 2(x - 1) - x(x + 1)
(xiii) 2x² - 4xy + 3y² - (8x² - 2xy + 6y²)
(xiv) (2x³ + 5x) - (3x² + 2x) + (x³ - 6x) - (x + 4x²)
(xv) 5ab + 6a²b - 7ab² + 2a²b - 3ab

Factorisation
3) Factorise...
(i) 4x + 8y
(ii) 12a + 15b - 18c
(iii) 72f - 26g - 48h
(iv) p² - pq + pr
(v) 12k² + 144km - 72kn
(vi) x² - 36x
(vii) 6x² - 36x
(viii) 2x² - 4x
(ix) 5x² - 2x
(x) x² + 3x
(xi) 4x² + 4x
(xii) b² - b³
(xiii) a² + ab
(xiv) 5a² - 10a
(xv) y⁵ - y⁴
(xvi) 18c³ - 9cd²
(xvii) 4a² - 16a²b

Multiplication
4) Multiply...
(i) 2xy × 3x²y
(ii) 5a²bc³ × 2ab² × 3c
(iii) km × mn × nk
(iv) 3pq²r × 6p²qr × 9pqr²
(v) rs × 2st × 3tu × 4ur
(vi) a × a × a
(vii) 4 × b × b
(viii) x³ × x⁵
(ix) y² × y² × y⁷
(x) 3b³ × 2b²
(xi) 2a × 3a² × 4a³
(xii) 3x² × 5x³y²

Fractions
5) Simplify...
(i) ^ab⁄_ac
(ii) ^2e⁄_4f
(iii) ^x²⁄_5x
(iv) ^4a²b⁄_2ab
(v) ^6p²q³r⁄_3p³q³r²
(vi) ^a⁄_b × ^b⁄_c × ^c⁄_a
(vii) ^3x⁄_2y × ^8y⁄_3z × ^5z⁄_4x
(viii) ^p²⁄_q × ^q²⁄_p
(ix) ^2fg⁄_16h × ^4gh²⁄_4fh × ^32fh³⁄_12f³
(x) ^kmn⁄_3n³ × ^6k²m³⁄_2k³m

6) Write as single fractions...
(i) ^x⁄₂ + ^x⁄₃
(ii) ^2x⁄₅ - ^x⁄₃ + ^3x⁄₄
(iii) ^3z⁄₈ + ^2z⁄₁₂ - ^5z⁄₂₄
(iv) ^2x⁄₂₃ - ^x⁄₄
(v) ^y⁄₂ - ^5y⁄₈ + ^4y⁄₅
(vi) ³⁄_x + ⁵⁄_x
(vii) ¹⁄_x + ¹⁄_y
(viii) ⁴⁄_x + ^x⁄_y
(ix) ^p⁄_q + ^q⁄_p
(x) ¹⁄_a - ¹⁄_b + ¹⁄_c
(xi) ^{x + 1}⁄₄ + ^{x - 1}⁄₂
(xii) ^2x⁄₃ - ^{x - 1}⁄₅
(xiii) ^{3x - 5}⁄₄ + ^{x - 7}⁄₆
(xiv) ^{3(2x + 1)}⁄₅ - ^{7(x - 2)}⁄₂
(xv) ^{4x + 1}⁄₈ + ^{7x - 3}⁄₁₂

10) Simplify...
(i) ^{x + 3}⁄_{2x + 6}
(ii) ^{6(2x + 1)²}⁄_{3(2x + 1)⁵}
(iii) ^{2x(y - 3)⁴}⁄_{8x²(y - 3)}
(iv) ^{6x - 12}⁄_{x - 2}
(v) ^{(3x + 2)²}⁄_6x × ^x⁴⁄_{6x + 4}