**Number Systems**

**Page: 26**

**1. Find:**

**(i)64 ^{1/2}**

**Solution:**

64^{1/2} = (8×8)^{1/2}

= (8^{2})^{½}

= 8^{1} [⸪2×1/2 = 2/2 =1]

= 8

**(ii)32 ^{1/5}**

**Solution:**

32^{1/5 }= (2^{5})^{1/5}

= (2^{5})^{⅕}

= 2^{1} [⸪5×1/5 = 1]

= 2

**(iii)125 ^{1/3}**

**Solution:**

(125)^{1/3} = (5×5×5)^{1/3}

= (5^{3})^{⅓}

= 5^{1} (3×1/3 = 3/3 = 1)

= 5

**2. Find:**

**(i) 9 ^{3/2}**

**Solution:**

9^{3/2} = (3×3)^{3/2}

= (3^{2})^{3/2}

= 3^{3} [⸪2×3/2 = 3]

=27

**(ii) 32 ^{2/5}**

**Solution:**

32^{2/5 }= (2×2×2×2×2)^{2/5}

= (2^{5})^{2⁄5}

= 2^{2} [⸪5×2/5= 2]

= 4

**(iii)16 ^{3/4}**

**Solution:**

16^{3/4 }= (2×2×2×2)^{3/4}

= (2^{4})^{3⁄4}

= 2^{3} [⸪4×3/4 = 3]

= 8

**(iv) 125 ^{-1/3}**

**Solution:**

125^{-1/3 }= (5×5×5)^{-1/3}

= (5^{3})^{-1⁄3}

= 5^{-1} [⸪3×-1/3 = -1]

= 1/5

**3. Simplify**:

**(i) 2 ^{2/3}×2^{1/5}**

**Solution:**

2^{2/3}×2^{1/5 }= 2^{(2/3)+(1/5)} [⸪Since, a^{m}×a^{n}=a^{m+n}____ Laws of exponents]

= 2^{13/15} [⸪2/3 + 1/5 = (2×5+3×1)/(3×5) = 13/15]

**(ii) (1/3 ^{3})^{7}**

**Solution:**

(1/3^{3})^{7 }= (3^{-3})^{7} [⸪Since,(a^{m})^{n }= a^{m x n}____ Laws of exponents]

= 3^{-21}

**(iii) 11 ^{1/2}/11^{1/4}**

**Solution:**

11^{1/2}/11^{1/4 }= 11^{(1/2)-(1/4)}

= 11^{1/4} [⸪(1/2) – (1/4) = (1×4-2×1)/(2×4) = 4-2)/8 = 2/8 = ¼ ]

**(iv) 7 ^{1/2}×8^{1/2}**

**Solution:**

7^{1/2}×8^{1/2} = (7×8)^{1/2} [⸪Since, (a^{m}×b^{m }= (a×b)^{m} ____ Laws of exponents]

= 56^{1/2}