Exponents and Power
Category : 7th Class
EXPONENTS AND POWER
FUNDAMENTALS
There are two part of an exponent.
Exponent \[\to \] base, Power/ Index
Example:
\[a\text{ }\times \text{ }a\text{ }\times \text{ }a={{a}^{3}}\] (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)
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
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) \[5{}^\circ \times 2{}^\circ \times 3{}^\circ =1\times 1\times 1=1\]
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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