90.1.4 Rational Number Part 1.4 Indix and power (Part 2)

90.1.4 Rational Number Part 1.4 Indix and power (Part 2)

Mind it

★ a⁰        = 1

★ a^m × aⁿ  = (a)^m+n

★ (a)^m+n=  a^m × aⁿ 

★ a^l × a^m × aⁿ  = (a)^{l + m +n}

★ (a^m) ⁿ = a^n×m

★ (a^m) ⁿ ] ^p = a^m×n×p

★ a^m × b^m  = (ab)^m

★ (ab)^m =  a^m × b^m 

★ a^m  ÷ aⁿ  = (a)^{m –n}

★ a^m  ÷ b^m  = (a/b)^m

★ (a/b)^m  = (b/a)^–n

Reciprocals of rational numbers:

Reciprocals of

 (i) a = 1/a

 (ii) a^m = 1/a^–m

 (iii) a^–m = 1/a^m

 (iv) a/b = b/a

 (v) (a/b)^m = (a/b)^–m

 (vi) (a/b)^–m = (a/b)^m

 (vii) (a/b)^m = (b/a)^m

 (ii) (b/a)^m = (a/b)^m

a) Converting the positive exponent into negative:

(i)

               1

a^m  =    _____

               a^–m

(ii)

(a/b)^m = (b/a)^–m     

b) Converting the negative exponent into posative:

(i)

                   1

a^–m  =     ______

                  a^m

(ii)

(a/b)^–m = (b/a)^m     

Identity 1

(a + b) (a + b) 

(a + b)² = a² + b² + 2ab 

                  Or

a² + b² + 2ab = (a + b)²

Identity 2

(a – b) (a – b) 

(a – b)² = a² + b^{2 –} 2ab 

                  Or

a² + b^{2 –} 2ab = (a – b)²

Identity 3

(a +b)(a –b)  = a² – b²  

                  Or

a² – b²  = (a +b)(a –b)

Identity 1

r₁²

01. Find the value of the followings:

(01) 3² × 3⁵ 

(02) 2^–5 × 2⁵ 

(03) 2⁵ × 2⁴

(04) 3² × 3⁵

(05) 5^–3 × 5⁵

(06) 2^–5 × 2^–5 

(07) 2² × 2³ × 2⁴ × 2⁵

(08) 2³ × 2⁵ × 2² × 2⁷ × 2⁹

(09) 3⁵ × 3⁴ × 3²

(10) 2¹ × 2⁹ × 2² × 2⁸ × 2³ × 2⁷ × 2⁵

(11) a^–5 × a^–5 

(12) m^–5 × m⁵ 

(13) p⁵ × p⁵

(14) x² × x⁵

(15) n^–3 × n⁵

02. Find the value of the followings:

(01) 3^1/2 × 3^1/5 

(02) 2^2/3 × 2^1/3 

(03) 2^1/5 × 2^1/5

(04) a^1/2 × a^1/5 

(05) 5^–3/4 × 5^5/4

(06) (3a)^2/5 × (3a)^3/5

(07) 2^1/2 × 2^3/2 × 2^4/2 × 2^5/2

(08) 2^1/4 × 2^2/4 × 2^3/4 × 2^5/4

(09) 3^5/6 × 3^4/6 × 3^2/6

(10) 2^1/7 × 2^4/7 × 2^2/7 × 2^3/7 × 2^6/7 × 2^5/7

02. Find the value of the followings:

(01) 3^1/2  ×  3^1/2 

(02) 2^2/3  ×  2^1/3 

(03) 2^1/5  ×  2^1/5

(04) a^1/5  ×  a^1/5 

(05) 5^–3/4  ×  5^5/4

(06) 3^1/2  ×  3^1/3 

(07) 2^2/3  ×  2^1/4 

(08) 2^1/5  ×  2^1/2

(09) a^1/2  ×  a^1/5 

(10) 5^–3/4  ×  5^5/7

(11) (3a)^2/5  ×  (3a)^3/4

(12) 2^1/2  ×  2^3/5

(13) 2^1/4  ×  2^2/5

(14) 3^5/6  ×  3^4/5

(15) 21/7  ÷ 24/5

3. Find the value of the followings:

Or

Solve the followings:

(01) 3² × 5² 

(02) 2³ × 5³ 

(03) 5⁵ × 2⁵

(04) a² × b² 

(05) 5^1/2 × 2^1/2

(06) (3a)^1/5 × (5b)^1/5

(07) 2^1/2 × 3^1/2 × 5^1/2

(08) 2^1/2 × 3^1/2 × 5^1/2 × 7^1/2

(09) 5^1/4 × 4^1/4 × 3^1/4 × 2^1/4

(10) 3^4/6 × 2^4/6 × 6^4/6

04. Find the value of the followings:

(01) 3⁷ ÷ 3⁵ 

(02) 5^–5 ÷ 5³   

(03) 2⁵ ÷ 2⁴

(04) 3² ÷ 3⁵

(05) 5^–3 ÷ 5⁵

(06) 6^–5 ÷ 6^–5 

(07) 2² ÷ 2³ 

(08) 7⁹ ÷ 7⁵

(09) 13⁵ ÷ 13⁴ 

(10) 12⁵ ÷ 12⁴ 

5. Find the value of the followings:

(01) 3^1/2  ÷ 3^1/2 

(02) 2^2/3  ÷ 2^1/3 

(03) 2^1/5  ÷ 2^1/5

(04) a^1/5  ÷ a^1/5 

(05) 5^–3/4  ÷ 5^5/4

(06) 3^1/2  ÷ 3^1/3 

(07) 2^2/3  ÷ 2^1/4 

(08) 2^1/5  ÷ 2^1/2

(09) a^1/2  ÷ a^1/5 

(10) 5^–3/4  ÷ 5^5/7

(11) (3a)^2/5  ÷ (3a)^3/4

(12) 2^1/2  ÷ 2^3/5

(13) 2^1/4  ÷ 2^2/5

(14) 3^5/6  ÷ 3^4/5

(15) 2^1/7  ÷ 2^4/5

6. Find the value of the followings:

(ii) 6¹⁵ ÷ 6¹⁰

(iii) a³ × a²

(iv) 7^x × 7²

(v) (5²)³ ÷ 5³

(vi) 2⁵ × 5⁵

(vii) a⁴ × b⁴

(viii) (3⁴)³

(x) 8^{t ÷} 8²

(v) 8² ÷ 2³

(iii) 25⁴ ÷ 5³

(vi) 2³ × a³ × 5a⁴

(vi) 2³ × a³ × 5a⁴

(vi) 2³ × a³ × 5a⁴

(i) 3² × 3⁴ × 3⁸

(ii) 2³ x 2² x 5⁵

(vii) 2⁰ × 3⁰ × 4⁰

(viii) (3⁰ + 2⁰ ) × 5⁰

(ix) (2²⁰ ÷ 2¹⁵) × 2³

(i) (3⁷ / 3² ) x 3⁵

(i) (3⁷ ÷ 3² ) x 3⁵

(iii) (6² x 6⁴) ÷ 6³

(iv) ((2²)³ x 3⁶ ) x 5⁶

(ii)[(5²)³ ×5⁴] ÷ 5⁷

(xii) (2³ × 2)²

7. Find the value of the followings:

(1) 

         3⁷ 

  __________

     3⁴ × 3³

(2) 

   3 × 7² × 11⁸

  ___________

     21 × 11³

(3)

   2³ × 3³ × 4

  ___________

      3 × 32

(4)

   a⁵ 

  ___ × a⁸

    a³

(5) 

     2⁸ × a⁵

  __________

     4³ × a³

(6) 

   4⁵ × a⁸b³

  ___________

     4⁵ × a⁵b²

(7) 

   (2⁵)² × 7³ 

  ___________

      8³ × 7

(8) 

   25 × 5² × t⁸

  ___________

     10³ × t⁴

(9) 

   3⁵ × 10⁵ × 25

  ___________

        5⁷ × 6⁵

(10) 

   12⁴ × 9³ × 4

  ______________

     6³ × 8² × 27

(11) 

   2³ × 3⁴ × 2⁵

  ___________

       9 × 4²

(12) 

   7⁴ × 7³ 

  ________

        7⁶

(13) 

        3⁸ 

  _________

     3⁴ × 9²

(14) 

           111⁶

     ___________

     111^–3 × 111²

(15) 

           3^1/2

     ___________

      3^1/4 × 9^1/2

(16) 

      7^1/4  × 7^3/4

     ____________

                7⁰ 

8. Find the value of the followings:

(1) 

(1² + 2² + 3 )²

(2) Find the value of p

5 ^{(p – 3)} × 5 ^{(2p – 8)} =  225

(3) 

[{(625)^–1/2 }^–2/3 ]²

(4) 

           4                        1

     _________   __    __________

        216^–2/3            216^–3/4