Note on Indices

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Which is the greatest number in the following?

.

(3)4, (9)3, (81)2

Here, the bases of all three terms and different. So it is difficult to compare. Therefore, converting all bases into the same base;

9 = 32

\(\therefore\) 9= (32)= 32×3 = 36

Similarly, 81 = 34

\(\therefore\) (81)= (34)= 34×2 = 38

Now, 3 is the base in 34, 36, 38. Therefore, the number having the greatest exponent with the same base is the greatest number.

∴ 38 or (81)2 is the greatest number.


Let us recall the laws of indices which we have studied in the previous classes.

1. Law of zero index: x0 = 1 

2. Product law of indices: xm \(\times\) xn = x m+n , (power are added in multiplication of same bases)

3. Power law of indices: (xm)n = x m\(\times\)h

4. Law of negative index: x-m = \(\frac{1}{x^m}\)

5. Root law of indices: x\(\frac pq\) = \(\sqrt[q]{x^p}\) = (\(\sqrt[q]{x}\))p

6. Quotient law of indices: a÷ an = \(\frac{a^m}{a^n}\) = \(\frac{1}{a^{(n-m)}}\)

(\(\frac{x}{y}\))n= \(\frac{x^n}{y^n}\)

7. (xy) m = xm ym

8. \(\sqrt[n]{x}\) = x\(\frac 1n\) 

These rules are known as laws of indices.

Example:

Find the value of: \(\sqrt[3]{8^2}\)

\(\sqrt[3]{8^2}\)

= 8\(\frac{2}{3}\)

= 23×\(\frac{2}{3}\)

= 22

= 4Ans

Exponential equation:

Exponential equation is an algebraic equation where unknown variables appears as an exponent of a base. We equate the power if the base on both sides of a equation is equal. We use the following rules while solving exponential equation:

1. If ax = ay then x = y

2. If ax = 1, then ax= a0

∴ x = 0

3. If ax = k1, then a = k\(\frac 1x\)

4. If ax = by, then a = b\(\frac yx\)

Some useful  formulas:

  • x0 = 1
  • xm \(\times\) xn = x m+n , (power are added in multiplication of same bases)
  • (xm)n = x m\(\times\)
  • (xy) m = xm ym
  • (\(\frac{x}{y}\))= \(\frac{x^n}{y^n}\)
  • x\(\frac pq\) = \(\sqrt[q]{x^p}\)=(\(\sqrt[q]{x}\))p
  • \(\sqrt[n]{x}\) = x\(\frac 1n\)
  • x-m = \(\frac{1}{x^m}\)
.

Very Short Questions

Soln:\(\sqrt{3}{8x^{-3}}\)

=\(\sqrt[3]\frac{8}{x^3}\)

= (\(\frac {8}{x^3}\))\(\frac 13\)

={ (\(\frac {2}{x}\))\(^3\)}\(\frac 13\)

= \(\frac{2}{x}\) Ans.

Solu:\({\left( \frac{5^{\frac{4}{3}}.5^{\frac{4}{3}}}{25^{\frac{1}{2}}} \right)}^\frac{3}{4}\)

= \({\left( \frac{5^{\frac{4}{3}}.-^{\frac{1}{3}}}{5^2{\frac{1}{2}}} \right)}^\frac{3}{4}\)

= \({\left( \frac{5^{\frac{4-1}{3}}}{5} \right)}^\frac{3}{4}\)

= \({\left( \frac{5^{\frac{3}{3}}}{5} \right)}^\frac{3}{4}\)

= \({\left( \frac{5^1}{5} \right)}^\frac{3}{4}\)

= (1)\(\frac{3}{4}\)

Soln: \({\left( \frac{x^0}{64} \right)}^\frac{-2}{3}\)

=\({\left( \frac{1}{64} \right)}^\frac{-2}{3}\)

=\({\left( \frac{1}{4^3} \right)}^\frac{-2}{3}\)

=\({\left(4^{-3} \right)}^\frac{-2}{3}\)

= (4)2

=16 Ans.

Soln:(\(\sqrt{3}\))6

=(\(\sqrt{3^\frac{1}{2}}\))6

= 33

= 27 Ans.

Soln: 4x=43

∴ x=3 Ans.

To verify: Put the value of x in given equation,

43=64 is true.

Soln:27x=3x+4

or, 33x=3x+4

or, 3x=x+4

or, 3x-x=4

or, 2x=4

∴ x=4/2= 2 Ans.

To verify: Put the value of x in given equation,

272=3x+4

or, 729= 36

or,729=729 is true.

Soln:92x=33-2x

or, (32)2x=33-2x

or, 34x=33-2x

or, 4x=3-2x

or, 4x+2x=3

or, 6x=3

∴ x=\(\frac{3}{6}\)=\(\frac{1}{2}\) Ans.

Now, putting the value of x in given equation for the verification.

92×\(\frac{1}{2}\)=33-2×\(\frac{1}{2}\)

or, 9=33-1

or, 9=32

or, 9=9 is true.

Soln: 2×83=2x-4

or, 2×(22)3 =2x-4

or, 21×29=2x-4

or, 210=2x-4

or,10=x-4

or, 10+4=x

∴x=14 Ans.

To verify: Put the value of x in given equation.

2×83=214-4

or, 2× (23)3=210

or, 21×29=210

or,210=210 is true.

Soln:2x+1-2x-8=0

or,2x+1-2x=8

or, 2x.2-2x=8

or,2x(2-1)=23

or, 2x.1=23

∴x=3 Ans.

To verify: Put the value of x in given equation

23+1-23-8=0

or, 24-8-8=0

or, 16-16=0

or, 0=0 is true.

Soln:2x+3.3x+4=18

or,2x×23×3x×34 =18

or,2x×8×3x×81 =18

or,2x×3x=\(\frac{18}{8×81}\)

or, (2×3)x=\(\frac{1}{36}\)

or, 6x=\(\frac{1}{6^2}\)

or, 6x=6-2

∴x=-2 Ans.

To verify: Put the value of x given equation

2-2+3.3-2+4=18

or, 21.32=18

or, 2.9=18

or, 18=18 is true.

Soln:22x-3.5x-1=200

or,22x×2-3×5x×5-1=200

or,22x×5x=\(\frac{200}{2^{-3}.5^{-1}}\)

or,4x.5x =200× 23× 5

or,(4×5)x=200×8×5

or, (20)x=8000

or, (20)x=8000

or,(20)x=(20)3

∴x=3 Ans.

To verify: Put the value of x given equation

26-3.33-1=200

or, 23.52=200

or, 8 25=200

or, 200=200 is true.

Soln:2x-4=4x-6

or,2x-4=22(x-6)

or,2x-4=22x-12

or,x-4=2x-12

or,x-2x=-12+4

or, -x=-8

∴x=8 Ans.

To verify: Put the value of x given equation

28-4=48-6

or, 24=42

or,16=16 is true

Here, 3x=27

or, 3x=33

∴ x=3

0%
  • 16-4÷32-4

    16


    13


    14


    15


  • (frac{14^6×15^5}{35^6×6^5})

    (frac{2}{5})


    (frac{2}{6})


    (frac{2}{7})


    (frac{2}{8})


  • (frac{12^7×28^6}{21^7×16^6})

    (frac{4}{8})


    (frac{4}{9})


    (frac{5}{7})


    (frac{4}{7})


  • (64x3÷27a-7)-(frac{2}{3})

    (frac{9}{16a^4x^2})


    (frac{9}{16a^6x^2})


    (frac{9}{16a^2x^2})


    (frac{9}{16a^2x^3})


  • (125a3÷27b-3)-(frac{2}{3})

    (frac{9}{25a^2b^5})


    (frac{9}{25a^2b^3})


    (frac{9}{25a^2b^4})


    (frac{9}{25a^2b^2})


  • (8x3÷27a-3)-(frac{2}{3})

    (frac{9}{4a^4y^2})


    (frac{9}{4a^5y^2})


    (frac{9}{4a^2y^2})


    (frac{9}{4a^3y^2})


  • (xa)b-c.(xb)c-a.(xc)a-b

    2


    6


    4


    1


  • (frac{5^{m+2}- 5^{m}} {5^{m+1} + 5^{m}})

    4


    2


    1


  • (frac{6^{n+2}- 6^{m}} {6^{n+1} + 6^{n}})

    6


    3


    4


    5


  • (frac{4^{m} + 4^{m+1}}{4^{m+2}- 4^{m}})

    (frac{1}{5}


    (frac{1}{2}


    (frac{1}{1}


    (frac{1}{3}


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gopi

pq^(x-3)=qp^(x-3)


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3^(x-2)=9a^(x-4)

please solve


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