## Exponents and Power

Category : 7th Class

EXPONENTS AND POWER

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.

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

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

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$a\text{ }\times \text{ }a\text{ }\times \text{ }a$....... (n factors) $=\text{ }{{a}^{n}}$(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., ${{(-5)}^{6}}=(-5)\times (-5)\times (-5)\times (-5)\times (-5)\times (-5)=15625$

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

e.g., ${{(-3)}^{5}}=(-3)\times (-3)\times (-3)\times (-3)\times (-3)=-243$

Note: (a) ${{(-1)}^{odd\,\,number}}=-1$                                  (b) ${{(-1)}^{even\,\,number}}=+1$

Elementary question 1:

In ${{3}^{5}},$what is the base and power respectively?

Ans.:    Base = 3

Power = 5

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)$a\times a\times a\times ~$............. $\times \text{ }a$(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) ${{({{a}^{m}})}^{n}}={{a}^{mn}}$

(e) ${{(ab)}^{m}}={{a}^{m}}{{b}^{m}}$

(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).

Elementary question 3:

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) ${{\left( \frac{5}{2} \right)}^{2}}$  (vii) $5{}^\circ \times 2{}^\circ \times 3{}^\circ$

Ans.:    (i) $5\times 5\times 5$(three times)$={{5}^{3}}=125$

(ii) ${{5}^{2}}\times {{5}^{3}}=\text{ }{{5}^{2+3}}=\text{ }{{5}^{5}}=\text{ }3125$

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

(iv) ${{({{5}^{2}})}^{3}}={{5}^{2\times 3}}={{5}^{6}}=15625$

(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) $5{}^\circ \times 2{}^\circ \times 3{}^\circ =1\times 1\times 1=1$

•                     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 $63.2=6.32\times 10=6.32\text{ }\times {{10}^{1}}$

Elementary question 4;

Write 2346 in standard form

Ans.:    $2.346\times 1000$so that decimal is after first non-zero digit$(2)=2.346\times {{10}^{3}}$.

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