**Exercise 2.1**

**Question 1. Simplify the following:-**

**(i) 3(a**

^{4}b^{3})10 × 5(a^{2}b^{2})^{3}**(ii) (2x**

^{-2}y^{3})^{3}**(iii) ((4 × 10**

^{7}) (6 × 10^{-5}))/(8 × 10^{4})**(iv) (4ab**

^{2}(-5ab^{3}))/(10a^{2}b^{2})**(v) {(x**

^{2}y^{2})/(a^{2}b^{2})}^{n}**(vi) (a**

^{3n-9}**)**

^{6}/a^{2n-4}**Solution 1.**

**(i)** We have 3(a^{4}b^{3})^{10} × 5(a^{2}b^{2})^{3}

By using identity:- (*x*^{m})^{n} = *x*^{mn}

After simplification we get

= 3(a^{40}b^{30}) × 5(a^{6}b^{6})

= 3 × a^{40 }× b^{30} × 5 × a^{6 }× b^{6}

By using identity:- *x*^{m }× *x*^{n} = *x*^{m+n}

= 15 × a^{46 }× b^{36}

= 15a^{46 }b^{36} **Answer.**

** **

**(ii) **(2*x *^{-2} *y*^{3})^{3}

By using identity:- (x^{-m})= 1/x^{m } and (*x*^{m})^{n} = *x*^{mn}

After simplification we get

= 2^{3} × ^{1}/x^{2x3} × *y*^{3×3}

= 8 × 1/x^{6} × *y*^{9}

= 8 *y*^{9}

= 8 *x ^{-6}*

*y*

^{6}

**Answer.**

**Question 2. If a = 3 and b = - 2, find the value of:**

**(i) a**

^{a}+ b^{b}**(ii) a**

^{b}+ b^{a}**(iii) (a+b)**

^{ab}**Solution 2.**

^{a}+ b

^{b}

^{3}+ (-2)

^{-2}

^{2}))

**= 109/4 Answer.**

^{b}+ b

^{a}

^{-2}+ (–2)

^{3}

^{2})) + (–2 × –2 × –2)

**= -71/9 Answer.**

^{ab}

^{3×(–2)}

^{–6}

^{–6}

^{–6}

^{6}

**= 1 Answer.**

**Question 3. Prove that:**

**Solution 3.**

**Question 4. Prove that:**

**Solution 4.**

**Question 5. Prove that:**

**Solution 5.**

**Question 8. Solve the following equation for x:**

**(i) 7**

^{(2x+3)}= 1**(ii) 2**

^{(x+1)}= 4^{(x-3)}**(iii) 2**

^{(5x+3)}= 8^{(x+3)}**(iv) 4**

^{2x}= 1/32**(v) 4**

^{(x-1)}×(0.5)^{(3-2x)}= (1/8)^{x}**(vi) 2**

^{(3x-7)}= 256**Solution 8.**

^{(2x+3)}= 1

^{0}=1

^{(2x+3)}= 7

^{0}

**x = (-3)/2 Answer.**

^{(x+1)}= 4

^{(x-3)}

^{m})

^{n}= x

^{mn}

^{(x+1)}= (2

^{2})

^{(x-3) }

^{(x+1)}= (2)

^{2(x-3)}

^{(x+1)}= (2)

^{(2x-6)}

**x = 7 Answer.**

^{(5x+3)}= 8

^{(x+3)}

^{m})

^{n}= x

^{mn}

^{(5x+3)}= (2

^{3})

^{(x+3)}

^{(5x+3)}=(2)

^{3(x+3)}

^{(5x+3)}=2

^{(3x+9) }

**x = 3 Answer.**

^{2x}= 1/32

^{2x}= 1/(2)

^{5 }

^{2})

^{2x}= 1/(2)

^{5 }

^{m})

^{n}= x

^{mn}

^{4x}= 2

^{(-5)}

**x = -5/4 Answer.**

^{(3x-7)}= 256

^{(3x-7)}= 2

^{8}

**x = 5 Answer.**

**Question 9. Solve the following equations for x:**

**(i) 2**

^{2x }**-2**

^{2x+3}+2^{4}= 0**(ii) 3**

^{(2x+4)}+1=2×3^{(x+2)}**Solution 9.**

**Question 10. If 49392 = a**

^{4}b^{2}c^{3}, find the value of a, b and c where a, b and c are different positive primes.**Solution 10.**

^{4}b

^{2}c

^{3}

**Hence the value of a = 2, b = 3 and c = 7 Answer.**

**Question 11. If 1176 = 2**

^{a}3^{b}7^{c}, find a,b and c.**Solution 11**

^{a}3

^{b}7

^{c}

^{3},3

^{1}and 7

^{2}.

**Hence the value of a = 3, b = 1 and c = 2 Answer.**

**Question 12. Given 4725 = 3**

^{a}5^{b}7^{c}, find**(i) the integral value of a, b and c**

**(ii) the value of 2-a3b7c**

**Solution 12.**

^{a}5

^{b}7

^{c}

^{3},5

^{2}and 7

^{1}.

**Exercise 2.2**

**Question 1. Assuming that x,y,z are positive real numbers, simplify each of the following:**

**Solution 1.**

**Exercise 2.3**

**Question 1. Write (625)**

^{-1/4}in decimal form.**Solution 1.**

^{-1/4}

^{4})

^{-1/4}

^{-4/4 }

^{-1}

**Question 2. State the product law of exponents.**

**Solution 2.**

^{m }× x

^{n }= x

^{(m+n)}

^{m }× x

^{n }=(x×x×x……………to m factors)×(x×x×x……………to n factors)

^{(m+n)}

^{m}×a

^{n}= a

^{(m+n)}

**Question 3. State the quotient law of exponents.**

**Solution 3.**

^{m}/x

^{n}

^{m}/x

^{n}=x

^{(m-n)}

**Question 4. State the power law of exponents.**

**Solution 4.**

^{m})

^{n }=x

^{m}× x

^{m}× x

^{m }× x

^{m}……………..n factors

^{m})

^{n }=(x×x×x×…..m)× (x×x×x×…..n)

^{m})

^{n }=(x×x×x…..mn) factors

^{m})

^{n }= x

^{mn }

**Question 5. If 2 ^{4}×4^{2}=16^{x}, then find the value of x.**

**Solution 5.**

We have 2^{4}×4^{2 }= 16^{x}

By the prime factorisation of 16 we get 2×2×2×2.

2^{4}×4^{2 }= (2^{4} )^{x}

2^{4}×(2^{2})^{2}=(2^{4})^{x }

2^{4 }× 2^{4}=(2^{4})^{x}

2^{8}=2^{4x}

By equating the exponents we get

8 = 4x

8/4=x

**2=x Answer**

**Question 6. If 3**

^{x-1}=9 and 4^{y+2}=64, what is the value of x/y?**Solution 6.**

^{x-1}=9

^{x-1 }= 3×3

^{x-1 }= 3

^{2}

^{y+2}= 64

^{y+2 }= 4×4×4

^{y+2 }= 4

^{3}

**x/y=3 Answer**

**Question 14. If (x-1)**

^{3}= 8, What is the value of (x+1)^{2 }?**Solution 14.**

^{3 }= 8

^{3 }= 2

^{3}

^{2}

^{2 }

^{2}

**=16 Answer**