Subject: Compulsory Maths
Indices are a useful way of more simply expressing large numbers. They also present us with many useful properties for manipulating them using what are called the Law of Indices.
Which is the greatest number in the following?
(3)4, (9)3, (81)2
Here, the bases of all three terms and different. So it is difficult to compare. Therefore, converting all bases into the same base;
9 = 32
\(\therefore\) 93 = (32)3 = 32×3 = 36
Similarly, 81 = 34
\(\therefore\) (81)2 = (34)2 = 34×2 = 38
Now, 3 is the base in 34, 36, 38. Therefore, the number having the greatest exponent with the same base is the greatest number.
∴ 38 or (81)2 is the greatest number.
Let us recall the laws of indices which we have studied in the previous classes.
1. Law of zero index: x0 = 1
2. Product law of indices: xm \(\times\) xn = x m+n , (power are added in multiplication of same bases)
3. Power law of indices: (xm)n = x m\(\times\)h
4. Law of negative index: x-m = \(\frac{1}{x^m}\)
5. Root law of indices: x\(\frac pq\) = \(\sqrt[q]{x^p}\) = (\(\sqrt[q]{x}\))p
6. Quotient law of indices: am ÷ an = \(\frac{a^m}{a^n}\) = \(\frac{1}{a^{(n-m)}}\)
(\(\frac{x}{y}\))n= \(\frac{x^n}{y^n}\)
7. (xy) m = xm ym
8. \(\sqrt[n]{x}\) = x\(\frac 1n\)
These rules are known as laws of indices.
Example:
Find the value of: \(\sqrt[3]{8^2}\)
\(\sqrt[3]{8^2}\)
= 8\(\frac{2}{3}\)
= 23×\(\frac{2}{3}\)
= 22
= 4Ans
Exponential equation is an algebraic equation where unknown variables appears as an exponent of a base. We equate the power if the base on both sides of a equation is equal. We use the following rules while solving exponential equation:
1. If ax = ay then x = y
2. If ax = 1, then ax= a0
∴ x = 0
3. If ax = k1, then a = k\(\frac 1x\)
4. If ax = by, then a = b\(\frac yx\)
Some useful formulas:
Find the value of:
\(\sqrt[3]{8x^{-3}}\)
Soln:\(\sqrt{3}{8x^{-3}}\)
=\(\sqrt[3]\frac{8}{x^3}\)
= (\(\frac {8}{x^3}\))\(\frac 13\)
={ (\(\frac {2}{x}\))\(^3\)}\(\frac 13\)
= \(\frac{2}{x}\) Ans.
Find the value of:
\({\left( \frac{5^{\frac{4}{3}}.5^{\frac{4}{3}}}{25^{\frac{1}{2}}} \right)}^\frac{3}{4}\)
Solu:\({\left( \frac{5^{\frac{4}{3}}.5^{\frac{4}{3}}}{25^{\frac{1}{2}}} \right)}^\frac{3}{4}\)
= \({\left( \frac{5^{\frac{4}{3}}.-^{\frac{1}{3}}}{5^2{\frac{1}{2}}} \right)}^\frac{3}{4}\)
= \({\left( \frac{5^{\frac{4-1}{3}}}{5} \right)}^\frac{3}{4}\)
= \({\left( \frac{5^{\frac{3}{3}}}{5} \right)}^\frac{3}{4}\)
= \({\left( \frac{5^1}{5} \right)}^\frac{3}{4}\)
= (1)\(\frac{3}{4}\)
Find the value of:
\({\left( \frac{x^0}{64} \right)}^\frac{-2}{3}\)
Soln: \({\left( \frac{x^0}{64} \right)}^\frac{-2}{3}\)
=\({\left( \frac{1}{64} \right)}^\frac{-2}{3}\)
=\({\left( \frac{1}{4^3} \right)}^\frac{-2}{3}\)
=\({\left(4^{-3} \right)}^\frac{-2}{3}\)
= (4)2
=16 Ans.
Find the values:
(\(\sqrt{3}\))6
Soln:(\(\sqrt{3}\))6
=(\(\sqrt{3^\frac{1}{2}}\))6
= 33
= 27 Ans.
Solve the following equation and verify them:
4x=64
Soln: 4x=43
∴ x=3 Ans.
To verify: Put the value of x in given equation,
43=64 is true.
Solve the following equation and verify them:
27x=3x+4
Soln:27x=3x+4
or, 33x=3x+4
or, 3x=x+4
or, 3x-x=4
or, 2x=4
∴ x=4/2= 2 Ans.
To verify: Put the value of x in given equation,
272=3x+4
or, 729= 36
or,729=729 is true.
Solve the following equation and verify them:
92x=33-2x
Soln:92x=33-2x
or, (32)2x=33-2x
or, 34x=33-2x
or, 4x=3-2x
or, 4x+2x=3
or, 6x=3
∴ x=\(\frac{3}{6}\)=\(\frac{1}{2}\) Ans.
Now, putting the value of x in given equation for the verification.
92×\(\frac{1}{2}\)=33-2×\(\frac{1}{2}\)
or, 9=33-1
or, 9=32
or, 9=9 is true.
Solve the following equation and verify them:
2×83=2x-4
Soln: 2×83=2x-4
or, 2×(22)3 =2x-4
or, 21×29=2x-4
or, 210=2x-4
or,10=x-4
or, 10+4=x
∴x=14 Ans.
To verify: Put the value of x in given equation.
2×83=214-4
or, 2× (23)3=210
or, 21×29=210
or,210=210 is true.
Solve the following equation and verify them:
2x+1-2x-8=0
Soln:2x+1-2x-8=0
or,2x+1-2x=8
or, 2x.2-2x=8
or,2x(2-1)=23
or, 2x.1=23
∴x=3 Ans.
To verify: Put the value of x in given equation
23+1-23-8=0
or, 24-8-8=0
or, 16-16=0
or, 0=0 is true.
Solve the following equation and verify them:
2x+3.3x+4=18
Soln:2x+3.3x+4=18
or,2x×23×3x×34 =18
or,2x×8×3x×81 =18
or,2x×3x=\(\frac{18}{8×81}\)
or, (2×3)x=\(\frac{1}{36}\)
or, 6x=\(\frac{1}{6^2}\)
or, 6x=6-2
∴x=-2 Ans.
To verify: Put the value of x given equation
2-2+3.3-2+4=18
or, 21.32=18
or, 2.9=18
or, 18=18 is true.
Solve the following equation and verify them:
22x-3.5x-1=200
Soln:22x-3.5x-1=200
or,22x×2-3×5x×5-1=200
or,22x×5x=\(\frac{200}{2^{-3}.5^{-1}}\)
or,4x.5x =200× 23× 5
or,(4×5)x=200×8×5
or, (20)x=8000
or, (20)x=8000
or,(20)x=(20)3
∴x=3 Ans.
To verify: Put the value of x given equation
26-3.33-1=200
or, 23.52=200
or, 8 25=200
or, 200=200 is true.
Solve the following equation and verify them:
2x-4=4x-6
Soln:2x-4=4x-6
or,2x-4=22(x-6)
or,2x-4=22x-12
or,x-4=2x-12
or,x-2x=-12+4
or, -x=-8
∴x=8 Ans.
To verify: Put the value of x given equation
28-4=48-6
or, 24=42
or,16=16 is true
Solve the following equation and verify them:
2x-4=4x-6
Soln:2x+2=\(\frac{2^{x+3}}{2}\)=1
or, \(\frac{2^{x+2.2^1+2^x+3}}{2}\)=1
or,2x+2+1.2x+3=2
or,2x+3.2x+3=2
or,2x+3(1+1)=2
or, 2x+3.2=2
or, 2x+3=\(\frac{2}{2}\)
or,2x+3=1
or,2x+3=0
∴x=-3 Ans.
To verify: Put the value of x given equation
2x+2+\(\frac{2^{x+3}}{2}\)=1
or,2-3+2+\(\frac{2^{-3+3}}{2}\)=1
or,2-1+\(\frac{2^0}{2}\)=1
or,\(\frac{1}{2}\)\(\frac{1}{2}\)=1 [∴2-1=\(\frac{1}{2}\) and 20=1]
or, \(\frac{1+1}{2}\)
or, \(\frac{2}{2}\)=1
or, 1=1 is true
Solve:3x=27
Here, 3x=27
or, 3x=33
∴ x=3
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