6th Class Mathematics Exponents and Power Notes - Exponent and Powers

Notes - Exponent and Powers

Category : 6th Class

EXPONENT AND POWERS

 

POWER

\[\frac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m}}-n\]

\[{{5}^{3}}\div {{5}^{2}}={{5}^{3}}-2\]

 

FUNDAMENTALS

  •                   Exponential form is nothing but repeated multiplication.

There are two part of an exponent.

Exponent\[\to \]base, Power/ Index

                                                           

Example:

           

  •                   Base denotes the number to be multiplied and the power denotes the number of times the base is to be multiplied.

\[a\times a\times a={{a}^{3}}\](read as 'a' cubed or 'a' raised to the power 3)

\[a\times a\times a\times a\times a\times a={{a}^{6}}\](read as 'a raised to the power 6 or 6th power of a)

            ...................................................................................

\[a\times a\times a\].......(n factors) \[={{a}^{n}}\] (read as 'a' raise to the power n or nth power of a)

  •                    (a) When a negative number is raised to an even power the value is always positive.

e.g., \[{{\left( -5 \right)}^{6}}=\left( -5 \right)\times \left( -5 \right)\times \left( -5 \right)\times \left( -5 \right)\times \left( -5 \right)\times \left( -5 \right)\]\[=15625\]

(b) When a negative number is raised to an odd power, the value is always negative.

e.g., \[{{\left( -3 \right)}^{5}}=\left( -3 \right)\times \left( -3 \right)\times \left( -3 \right)\times \left( -3 \right)\times \left( -3 \right)=\left( -243 \right)\]

Note:    (a) \[{{(-1)}^{odd\,\,number}}=-1\]

(b) \[{{(-1)}^{even\,\,number}}=+1\]

 

Elementary Question 2:

Write 32 in exponent form

Ans.     \[32=2\times 2\times 2\times 2\times 2={{2}^{5}}\]where base\[=2\]

            power / Index = 5

 

  •                   Laws of Exponents:

For any non-zero integers 'a' and 'b' and whole numbers 'm? and 'n',

(a)\[a\times a\times a\times \]............. \[\times a\](m factors) \[={{a}^{m}}\]

(b) \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]

(c) \[\frac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}},\]if \[m>n;=1,\] if \[m=n;\,\,=\frac{1}{{{a}^{n-m}}}\] if \[m<n\]

(d) \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}\]

(e) \[{{\left( ab \right)}^{m}}={{a}^{m}}{{b}^{m}}\]

(f) \[{{\left( \frac{a}{b} \right)}^{m}}=\frac{{{a}^{m}}}{{{b}^{m}}}\]

(g) \[a{}^\circ =1\]

Most of the questions under this chapter are applications of the above formula (a) to (g). Therefore commit them to memory (not ROT memory but learn by applying).

Evaluate:          (i) \[5\times 5\times 5\]     (ii) \[{{5}^{2}}\times {{5}^{3}}\]          (iii) \[\frac{{{5}^{3}}}{{{5}^{2}}}\]        (iv)\[{{\left( {{5}^{2}} \right)}^{3}}\]  

(v) \[{{\left( 2\times 5 \right)}^{3}}\]       (vi) \[{{\left( \frac{5}{2} \right)}^{1}};\]              (vii) \[5{}^\circ \times 2{}^\circ \times 3{}^\circ \]

Answer: (i) \[5\times 5\times 5\](three times) \[={{5}^{3}}=125\]

(ii) \[{{5}^{2}}\times {{5}^{3}}={{5}^{2+3}}={{5}^{5}}=3125\]

(iii) \[\frac{{{5}^{3}}}{{{5}^{2}}}={{5}^{3-2}}={{5}^{1}}=5\]

(iv) \[{{\left( {{5}^{2}} \right)}^{3}}={{5}^{2\times 3}}={{5}^{6}}=15625\]

(v) \[{{\left( \frac{5}{2} \right)}^{2}}=\frac{{{5}^{2}}}{{{2}^{2}}}=\frac{25}{4};\]

(vi) \[{{\left( 2\times 5 \right)}^{3}}={{2}^{3}}\times {{5}^{3}}=8\times 125=1000\]

(vii) \[{{5}^{0}}\times {{2}^{0}}\times {{3}^{0}}=1\times 1\times 1=1\]

  •                 Any number can be expressed as a decimal number between \[1.0\] and \[10.0\] including \[1.0\]multiplied by a power of 10. Such a form of a number is called its standard form.

For example, standard form of \[63.2\]\[=6.32\times 10\]\[=6.32\times {{10}^{1}}\]

 

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Notes - Exponent and Powers
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