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Hello friends, we usually encounter problems on surds and indices in competitive exams. So lets study and clear our doubts. Lets learn some basic properties of indices:

<ul>
	<li>am = a × a × a × a ×…..(m times)</li>
	<li>a0 = 1 </li>
	<li>am × an = am+n</li>
	<li>a m/ an = am-n</li>
	<li>(am) n = am × n</li>
	<li>a1/m = m√a</li>
	<li>an/m = m√a n</li>
</ul>

Note: If the power of any number is negative, the reciprocal of the same number makes the power positive.

Let us see some cases:

(a)-m = 1/(a)m

1/(a)-m = (a)m

(a)-m/(b)-n = (b)n/(a)m

(a/b)-m = (b/a)m

Let’s solve some problems on these properties.

E.g. Find the value of y, if 37 × 82 = 92 × 42 × 12y

Sol: We know that 9 = 32 , 4 = 22 and 8 = 23 then,

3 7 × (23 ) 2 = (32 ) 2 × (22 ) 2 × 12y

3 7 × 26 = 34 × 24 × 12y

3 3 × 22 = 12y

27 × 4/12 = y

y = 9 (Ans.)

E.g. Find the value of y, if

(27/64)2 × (3/4)2 × (9/16)2 × y + 1 = (81/256)3 × y

Sol: Now, (27/64)2 × (3/4)2 × (9/16)2 × y + 1 = (81/256)3 × y

(3/4)3 × 2 × (3/4)2 × (3/4)2 × (2 × y + 1) = (3/4)4 × 3 × y

(3/4)6 + 2 + 2 + 4 × y = (3/4)12 × y

(3/4)10 + 4 × y = (3/4)12 × y

10 + 4 × y = 12 × y

8 × y = 10 

y = 10/8 = 1.25 (Ans.)

Now, let’s learn some vividly used powers of numbers.

<h2>1. Squares</h2>

Squares refer to the numbers which are obtained when a number is multiplied by itself or raised to the power 2 i.e. Square of a = a × a = a2. The table below shows the squares of numbers from 1 to 30.

<table cellspacing="0" class="Table" style="border-collapse:collapse; border:none; width:300px">
	<tbody>
		<tr>
			<td>
			1
			</td>
			<td>
			121
			</td>
			<td>
			441
			</td>
		</tr>
		<tr>
			<td>
			4
			</td>
			<td>
			144
			</td>
			<td>
			484
			</td>
		</tr>
		<tr>
			<td>
			9
			</td>
			<td>
			169
			</td>
			<td>
			529
			</td>
		</tr>
		<tr>
			<td>
			16
			</td>
			<td>
			196
			</td>
			<td>
			576
			</td>
		</tr>
		<tr>
			<td>
			25
			</td>
			<td>
			225
			</td>
			<td>
			625
			</td>
		</tr>
		<tr>
			<td>
			36
			</td>
			<td>
			256
			</td>
			<td>
			676
			</td>
		</tr>
		<tr>
			<td>
			49
			</td>
			<td>
			289
			</td>
			<td>
			729
			</td>
		</tr>
		<tr>
			<td>
			64
			</td>
			<td>
			324
			</td>
			<td>
			784
			</td>
		</tr>
		<tr>
			<td>
			81
			</td>
			<td>
			361
			</td>
			<td>
			841
			</td>
		</tr>
		<tr>
			<td>
			100
			</td>
			<td>
			400
			</td>
			<td>
			900
			</td>
		</tr>
	</tbody>
</table>

<h2>2. Cubes</h2>

Cubes refers to the numbers which are obtained when a number is multiplied by itself thrice or raised to the power 3 i.e. Cube of a = a × a × a = a3

The table given below shows the cubes of numbers from 1 to 20.

<table cellspacing="0" style="width:300px">
	<tbody>
		<tr>
			<td>
			1
			</td>
			<td>
			1331
			</td>
		</tr>
		<tr>
			<td>
			8
			</td>
			<td>
			1728
			</td>
		</tr>
		<tr>
			<td>
			27
			</td>
			<td>
			2197
			</td>
		</tr>
		<tr>
			<td>
			64
			</td>
			<td>
			2744
			</td>
		</tr>
		<tr>
			<td>
			125
			</td>
			<td>
			3375
			</td>
		</tr>
		<tr>
			<td>
			216
			</td>
			<td>
			4096
			</td>
		</tr>
		<tr>
			<td>
			343
			</td>
			<td>
			4913
			</td>
		</tr>
		<tr>
			<td>
			512
			</td>
			<td>
			5832
			</td>
		</tr>
		<tr>
			<td>
			729
			</td>
			<td>
			6859
			</td>
		</tr>
		<tr>
			<td>
			1000
			</td>
			<td>
			8000
			</td>
		</tr>
	</tbody>
</table>

 

Hope this blog clear your concept. For more such blogs, stay connected with us. 

Thank you

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