# 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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