Math Mantra Nepal – Educational Portal
Day – 3: Indices, Factorization, HCF & LCM
1. Exponents and Laws of Indices
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a) If \( 4^x = 8^2 \), find the value of x.
\( 4^x = 8^2 \) or, \( 4^x = 64 \) [since \( 8^2 = 64 \)] or, \( 4^x = 4^3 \) [since \( 4^3 = 64 \)] \( \mathbf{x = 3} \) [equating powers with same base] -
b) What should be the power of 5, so that the value becomes \( \frac{1}{625} \)?
Let the power be \( x \). \( 5^x = \frac{1}{625} \) or, \( 5^x = \frac{1}{5^4} \) [since \( 5^4 = 625 \)] or, \( 5^x = 5^{-4} \) [using \( \frac{1}{a^n} = a^{-n} \)] \( \mathbf{x = -4} \) -
c) Simplify: \( (a^2b) \times (ab^3) \)
\( = a^2 \cdot a^1 \cdot b^1 \cdot b^3 \) \( = a^{2+1} \cdot b^{1+3} \) [using \( a^m \cdot a^n = a^{m+n} \)] \( \mathbf{= a^3b^4} \) -
d) Simplify: \( (x^6y^3) \div (x^3y^3) \)
\( = \frac{x^6y^3}{x^3y^3} \) \( = x^{6-3} \cdot y^{3-3} \) [using \( \frac{a^m}{a^n} = a^{m-n} \)] \( = x^3 \cdot y^0 \) \( \mathbf{= x^3} \) [since \( y^0 = 1 \)] -
e) Simplify: \( x^4 \div (2x^3) \)
\( = \frac{x^4}{2x^3} \) \( = \frac{1}{2} \cdot x^{4-3} \) [subtracting powers of same base] \( \mathbf{= \frac{1}{2}x} \) -
f) Simplify:
(i) \( \left(\frac{x^2}{y^2}\right)^2 \)
(ii) \( \left(\frac{64}{125}\right)^{\frac{1}{3}} \)
(i) \( \left(\frac{x^2}{y^2}\right)^2 = \frac{(x^2)^2}{(y^2)^2} = \frac{x^{2 \times 2}}{y^{2 \times 2}} \mathbf{= \frac{x^4}{y^4}} \) (ii) \( \left(\frac{64}{125}\right)^{\frac{1}{3}} = \left(\frac{4^3}{5^3}\right)^{\frac{1}{3}} = \left(\frac{4}{5}\right)^{3 \times \frac{1}{3}} \mathbf{= \frac{4}{5}} \) -
g) Simplify: \( \frac{a^{4n-2}}{a^{2(2n-1)}} \)
\( = \frac{a^{4n-2}}{a^{4n-2}} \) [opening bracket in denominator: \( 2 \times 2n – 2 \times 1 = 4n – 2 \)] \( = a^{(4n-2) – (4n-2)} \) [applying division law] \( = a^{4n – 2 – 4n + 2} \) [cancelling \( +4n, -4n \) and \( -2, +2 \)] \( = a^0 \mathbf{= 1} \) -
h) Evaluate: \( \frac{4^4 \times 5^5}{25^3 \times 16^2} \)
\( = \frac{(2^2)^4 \times 5^5}{(5^2)^3 \times (2^4)^2} \) \( = \frac{2^8 \times 5^5}{5^6 \times 2^8} \) \( = 2^{8-8} \times 5^{5-6} \) \( = 1 \times 5^{-1} \mathbf{= \frac{1}{5}} \) -
i) Evaluate: \( \frac{5^3 \times 125^3}{25^3} \)
\( = \frac{5^3 \times (5^3)^3}{(5^2)^3} \) \( = \frac{5^3 \times 5^9}{5^6} \) \( = \frac{5^{3+9}}{5^6} = \frac{5^{12}}{5^6} \) \( = 5^{12-6} = 5^6 \mathbf{= 15625} \) -
j) Simplify: \( (x^{a-b})^{a+b} \times (x^{b-c})^{b+c} \times (x^{c-a})^{c+a} \)
\( = x^{(a-b)(a+b)} \times x^{(b-c)(b+c)} \times x^{(c-a)(c+a)} \) \( = x^{a^2-b^2} \times x^{b^2-c^2} \times x^{c^2-a^2} \) [using formula \( (a-b)(a+b) = a^2-b^2 \)] \( = x^{(a^2 – b^2 + b^2 – c^2 + c^2 – a^2)} \) [adding powers since bases are same] \( = x^{(a^2 – \underline{a^2} + b^2 – \underline{b^2} + c^2 – \underline{c^2})} \) [cancelling \( +a^2, -a^2 \), \( +b^2, -b^2 \), \( +c^2, -c^2 \)] \( = x^0 \mathbf{= 1} \) -
k) Simplify: \( x^{a(b-c)} \times x^{b(c-a)} \times x^{c(a-b)} \)
\( = x^{ab – ac} \times x^{bc – ab} \times x^{ac – bc} \) \( = x^{ab – ac + bc – ab + ac – bc} \) \( = x^{(ab – ab) + (bc – bc) + (ac – ac)} \) [cancelling all terms] \( = x^0 \mathbf{= 1} \) -
l) Simplify: \( \{x^{(b-c)}\}^4 \times \{x^{(c-a)}\}^4 \times \{x^{(a-b)}\}^4 \)
\( = x^{4(b-c)} \times x^{4(c-a)} \times x^{4(a-b)} \) \( = x^{4b – 4c + 4c – 4a + 4a – 4b} \) \( = x^0 \mathbf{= 1} \)
Extra Practice Question for Students:
Simplify: \( \{x^{(b-c)}\}^4 \times \{x^{(a+c)}\}^4 \times \{x^{(b-a)}\}^4 \)
[Hint: Expand powers and simplify using laws of indices] -
m) Simplify: \( \frac{x^{m+n+2} \times x^{m+n+2}}{x^{2(m+n+1)}} \)
\( = \frac{x^{m+n+2} \cdot x^{m+n+2}}{x^{2(m+n+1)}} \) [Step 1: Expressing numerator terms clearly] \( = \frac{x^{(m+n+2) + (m+n+2)}}{x^{2m + 2n + 2}} \) [Step 2: Adding powers in numerator & multiplying denominator by 2] \( = \frac{x^{2m + 2n + 4}}{x^{2m + 2n + 2}} \) [Step 3: Combining like terms in power] \( = x^{(2m + 2n + 4) – (2m + 2n + 2)} \) [subtracting denominator power] \( = x^{2m + 2n + 4 – 2m – 2n – 2} \) [cancelling \( +2m, -2m \) and \( +2n, -2n \)] \( = x^{4 – 2} \) \( \mathbf{= x^2} \)
2. Factorization (Algebra)
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a) Factorize: \( x^3 + x^2 + x \)
\( \mathbf{= x(x^2 + x + 1)} \) -
b) Factorize: \( ax + bx – ay – by \)
\( = x(a + b) – y(a + b) \) \( \mathbf{= (a + b)(x – y)} \) -
c) Factorize: \( 2x – x^2 + 2 – x \) (Corrected)
\( = x(2 – x) + 1(2 – x) \) [rearranging/grouping] \( \mathbf{= (2 – x)(x + 1)} \) -
d) Factorize: \( 9x^2 – y^2 \)
\( = (\underline{3x})^2 – (\underline{y})^2 \) [expressing in \( a^2 – b^2 \) form] \( \mathbf{= (3x + y)(3x – y)} \) -
e) Factorize: \( 5x^2 – 20y^2 \)
\( = 5(x^2 – 4y^2) \) [taking common 5] \( = 5[(\underline{x})^2 – (\underline{2y})^2] \) \( \mathbf{= 5(x + 2y)(x – 2y)} \) -
f) Factorize: \( 13a^2 – 117b^2 \)
\( = 13(a^2 – 9b^2) \) [taking common 13] \( = 13[(\underline{a})^2 – (\underline{3b})^2] \) \( \mathbf{= 13(a + 3b)(a – 3b)} \) -
g) Factorize: \( 72 – 2b^2 \)
\( = 2(36 – b^2) \) [taking common 2] \( = 2[(\underline{6})^2 – (\underline{b})^2] \) \( \mathbf{= 2(6 + b)(6 – b)} \) -
h) Factorize: \( 25 – 16y^2 \)
\( = (\underline{5})^2 – (\underline{4y})^2 \) [expressing in \( a^2 – b^2 \) form] \( \mathbf{= (5 + 4y)(5 – 4y)} \) -
i) Factorize: \( 256 – x^4 \)
\( = (\underline{16})^2 – (\underline{x^2})^2 \) \( = (16 + x^2)(16 – x^2) \) \( = (16 + x^2)[(\underline{4})^2 – (\underline{x})^2] \) \( \mathbf{= (16 + x^2)(4 + x)(4 – x)} \) -
j) Factorize: \( a^2 + 12a + 36 \)
\( a^2 + \underline{2(a)(6)} + 6^2 \)Formula: \( a^2 + \mathbf{2ab} + b^2 = (a+b)^2 \)\( \mathbf{= (a + 6)^2} \) -
k) Factorize: \( 9r^2 + 60r + 100 \)
\( (3r)^2 + \underline{2(3r)(10)} + 10^2 \)Formula: \( a^2 + \mathbf{2ab} + b^2 = (a+b)^2 \)\( \mathbf{= (3r + 10)^2} \) -
l) Factorize: \( 49r^2 – 70r + 25 \)
\( (7r)^2 – \underline{2(7r)(5)} + 5^2 \)Formula: \( a^2 – \mathbf{2ab} + b^2 = (a-b)^2 \)\( \mathbf{= (7r – 5)^2} \) -
m) Factorize: \( x^3 – 10x^2 + 16x \)
\( = x(x^2 – 10x + 16) \) [taking common x] \( = x(x^2 – 8x – 2x + 16) \) [mid-term splitting] \( = x[x(x – 8) – 2(x – 8)] \) \( \mathbf{= x(x – 8)(x – 2)} \) -
n) Factorize: \( x^3 – 6x^2 + 8x \)
\( = x(x^2 – 6x + 8) \) [taking common x] \( = x(x^2 – 4x – 2x + 8) \) [mid-term splitting] \( = x[x(x – 4) – 2(x – 4)] \) \( \mathbf{= x(x – 4)(x – 2)} \) -
o) Factorize: \( x^3 – 3x^2 – 10x \)
\( = x(x^2 – 3x – 10) \) [taking common x] \( = x(x^2 – 5x + 2x – 10) \) [mid-term splitting] \( = x[x(x – 5) + 2(x – 5)] \) \( \mathbf{= x(x – 5)(x + 2)} \) -
p) Factorize: \( 25a^2 – 40ab + 16b^2 \)
\( (5a)^2 – \underline{2(5a)(4b)} + (4b)^2 \)Formula: \( a^2 – \mathbf{2ab} + b^2 = (a-b)^2 \)\( \mathbf{= (5a – 4b)^2} \) -
q) Factorize: \( a^2 + 12a + 36 – b^2 \)
\( = (a^2 + \mathbf{2 \times a \times 6} + 6^2) – b^2 \) \( = (a + 6)^2 – b^2 \) \( \mathbf{= (a + 6 + b)(a + 6 – b)} \) [using difference of squares] -
r) Factorize: \( 9r^2 – s^2 – 6s – 9 \)
\( = (3r)^2 – (s^2 + \mathbf{2 \times s \times 3} + 3^2) \) \( = (3r)^2 – (s + 3)^2 \) \( \mathbf{= (3r + s + 3)(3r – s – 3)} \) -
s) Factorize: \( x^2 + x – 20 \)
\( = x^2 + 5x – 4x – 20 \) [mid-term splitting] \( = x(x + 5) – 4(x + 5) \) \( \mathbf{= (x + 5)(x – 4)} \)
Interactive Tile-Based Factorization (Algebra Tiles)
Instructions: Drag tiles to move them. Double-click any tile to rotate 90° (swaps between Horizontal and Vertical orientations). Click ‘Start Auto-Arrange Animation’ to automatically form the square/rectangle and view factors!
a) Arrange into square form for \( x^2 + 4x + 4 \) (All positive: 1 \( x^2 \), 4 \( x \), 4 units)
\( x^2 \)
\( +x \)
\( +x \)
\( +x \)
\( +x \)
+1
+1
+1
+1
b) Arrange tiles for \( x^2 – 3x + 2 \) (1 positive \( x^2 \), 3 negative \( x \) (red), 2 positive units):
\( x^2 \)
\( -x \)
\( -x \)
\( -x \)
+1
+1
3. HCF, LCM & Special Factorization
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a) Find the HCF and LCM of \( 9x^2y^3 \) and \( 15xy^2 \)
First Expression: \( 9x^2y^3 = 3 \times \underline{3} \times \underline{x} \times x \times \underline{y \times y} \times y \) Second Expression: \( 15xy^2 = \underline{3} \times \underline{5} \times \underline{x} \times \underline{y \times y} \) Common Factors = \( \underline{3 \times x \times y \times y} = 3xy^2 \) Remaining Factors = \( 3 \times x \times 5 = 15xy \) \( \mathbf{HCF} = \text{Common Factor} \mathbf{= 3xy^2} \) \( \mathbf{LCM} = \text{Common Factor} \times \text{Remaining Factors} = 3xy^2 \times 15xy \mathbf{= 45x^2y^3} \) -
b) Find the HCF and LCM of \( (x^2 + 7x + 10) \) and \( (x^2 – x – 6) \)
First Expression: \( x^2 + 7x + 10 = x^2 + 5x + 2x + 10 = x(x+5) + 2(x+5) = (\underline{x+2})(x+5) \) Second Expression: \( x^2 – x – 6 = x^2 – 3x + 2x – 6 = x(x-3) + 2(x-3) = (\underline{x+2})(x-3) \) Common Factor = \( \underline{x+2} \) Remaining Factors = \( (x+5)(x-3) \) \( \mathbf{HCF} = \text{Common Factor} \mathbf{= (x + 2)} \) \( \mathbf{LCM} = \text{Common Factor} \times \text{Remaining Factors} \mathbf{= (x + 2)(x + 5)(x – 3)} \) -
c) Find the HCF and LCM of: \( a^2 – 2ab + b^2 \) and \( a^3 – ab^2 \)
First Expression: \( a^2 – 2ab + b^2 = (\underline{a – b})(a – b) \) Second Expression: \( a(a^2 – b^2) = a(\underline{a – b})(a + b) \) Common Factor = \( \underline{a – b} \) Remaining Factors = \( (a – b) \times a \times (a + b) \) \( \mathbf{HCF} = \text{Common Factor} \mathbf{= (a – b)} \) \( \mathbf{LCM} = \text{Common Factor} \times \text{Remaining Factors} \mathbf{= (a – b) \times (a – b) \times a \times (a + b)} \) -
d) Find the HCF and LCM of: \( x^2 + 6x + 8 \), \( x^2 – 4 \), and \( x^2 + 4x + 4 \)
First Expression: \( x^2 + 6x + 8 = (\underline{x + 2})(x + 4) \) Second Expression: \( x^2 – 4 = (\underline{x + 2})(x – 2) \) Third Expression: \( x^2 + 4x + 4 = (\underline{x + 2})(x + 2) \) Common Factor = \( \underline{x + 2} \) Remaining Factors = \( (x + 4) \times (x – 2) \times (x + 2) \) \( \mathbf{HCF} = \text{Common Factor} \mathbf{= (x + 2)} \) \( \mathbf{LCM} = \text{Common Factor} \times \text{Remaining Factors} \mathbf{= (x + 2) \times (x + 4) \times (x – 2) \times (x + 2)} \) -
e) Find the HCF and LCM of: \( x^2 + 5x + 6 \), \( x^2 – 9 \), and \( x^2 + 6x + 9 \)
First Expression: \( x^2 + 5x + 6 = (\underline{x + 3})(x + 2) \) Second Expression: \( x^2 – 9 = (\underline{x + 3})(x – 3) \) Third Expression: \( x^2 + 6x + 9 = (\underline{x + 3})(x + 3) \) Common Factor = \( \underline{x + 3} \) Remaining Factors = \( (x + 2) \times (x – 3) \times (x + 3) \) \( \mathbf{HCF} = \text{Common Factor} \mathbf{= (x + 3)} \) \( \mathbf{LCM} = \text{Common Factor} \times \text{Remaining Factors} \mathbf{= (x + 3) \times (x + 2) \times (x – 3) \times (x + 3)} \) -
f) Find the HCF and LCM of: \( 3x^3 – 15x^2 \) and \( 2x^3 – 50x \)
First Expression: \( 3x^3 – 15x^2 = 3 \times \underline{x} \times x \times (\underline{x – 5}) \) Second Expression: \( 2x^3 – 50x = 2 \times \underline{x} \times (\underline{x – 5})(x + 5) \) Common Factors = \( \underline{x(x – 5)} \) Remaining Factors = \( 3x \times 2(x + 5) = 6x(x + 5) \) \( \mathbf{HCF} = \text{Common Factor} \mathbf{= x(x – 5)} \) \( \mathbf{LCM} = \text{Common Factor} \times \text{Remaining Factors} \mathbf{= x(x – 5) \times 6x(x + 5)} \) -
g) Find the HCF and LCM of: \( x^2 + x – 20 \) and \( x^2 – 25 \)
First Expression: \( x^2 + x – 20 = (\underline{x + 5})(x – 4) \) Second Expression: \( x^2 – 25 = (\underline{x + 5})(x – 5) \) Common Factor = \( \underline{x + 5} \) Remaining Factors = \( (x – 4) \times (x – 5) \) \( \mathbf{HCF} = \text{Common Factor} \mathbf{= (x + 5)} \) \( \mathbf{LCM} = \text{Common Factor} \times \text{Remaining Factors} \mathbf{= (x + 5) \times (x – 4) \times (x – 5)} \) -
h) Find the HCF and LCM of: \( x^3 – x^2 – 42x \) and \( x^4 + 4x^3 – 12x^2 \)
First Expression: \( x^3 – x^2 – 42x = \underline{x}(\underline{x + 6})(x – 7) \) Second Expression: \( x^4 + 4x^3 – 12x^2 = \underline{x} \times x \times (\underline{x + 6})(x – 2) \) Common Factors = \( \underline{x(x + 6)} \) Remaining Factors = \( (x – 7) \times x \times (x – 2) \) \( \mathbf{HCF} = \text{Common Factor} \mathbf{= x(x + 6)} \) \( \mathbf{LCM} = \text{Common Factor} \times \text{Remaining Factors} \mathbf{= x(x + 6) \times (x – 7) \times x \times (x – 2)} \)