MATH EXPONENTS - NUMBER OF SOLVED EXAMPLES AND EXERCISES ON EXPONENTS

Please study  Laws of Exponents before Math Exponents,
if you have not already done so.

It is a prerequisite here. There, we stated the 7 laws of indices
and the two Rules used for solving problems.

We also gave the explanations and proofs of the 7 laws.

We provided a few Solved Examples and Problems for Practice with
Answers to help to apply the 7 Laws and the 2 Rules in solving problems.

Here, we provide many more Solved Examples and Exercises with answers.

Studying the worked out problems will help remember and apply
the 7 Laws of exponents and the 2 Rules.

Practice makes one perfect. This is especially true for
remembering Algebra Formulas (Math Formulas).
So, take the exercises seriously and practice solving the problems.

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Set of Solved Examples of Math Exponents :
Application of Laws of Exponents

Solved Example 1 of Math Exponents

Evaluate (-1⁄2)⁵⁄(-1⁄2)⁴ + (-1⁄8)⁄(-1⁄4)

Solution to Example 1 of Math Exponents:

We know a^m⁄aⁿ = a^{m - n}
Applying this Law, we get
(-1⁄2)⁵⁄(-1⁄2)⁴ + (-1⁄8)⁄(-1⁄4)
= (-1⁄2)^{5 - 4} + (-1⁄8) x (4⁄-1)
= (-1⁄2)¹ + 1⁄2
= -1⁄2 + 1⁄2 = 0. Ans.

Solved Example 2 of Math Exponents

What is (-1)⁴⁰¹?

Solution to Example 2 of Math Exponents:

Let A = (-1)⁴⁰¹
A = (-1)⁴⁰¹ = (-1) ^{400 + 1}
Interchanging L.H.S. and R.H.S. in Law 1, we have
a^{m + n} = a^m x aⁿ
Applying this here, we get
A = (-1)⁴⁰⁰ x (-1)¹
= (-1)^{2 x 200} x (-1)¹
Interchanging L.H.S. and R.H.S. in Law 2, we have
a^mn = (a^m)ⁿ
Applying this here, we get
A = {(-1)²}²⁰⁰ x (-1)¹
We know (-1)² = (-1) x (-1) = 1.
and (-1)¹ = -1.
∴ A = 1²⁰⁰ x (-1 )
We know 1power any number is 1.
∴ A = 1 x (-1 ) = -1.
Thus (-1)⁴⁰¹ = -1. Ans.

You might have noted (-1)^{odd number} = -1 and (-1)^{even number} = 1.

Solved Example 3 of Math Exponents

Given that 5^x = 1000, find 5^{x + 2} and 5^{x - 2}

Solution to Example 3 of Math Exponents:

Let A = 5^{x + 2} and B = 5^{x - 2}
Interchanging L.H.S. and R.H.S. in Law 1, we have
a^{m + n} = a^m x aⁿ
Applying this here, we get
A = 5^{x + 2} = 5^x x 5²
By data 5^x = 1000. Using this here, we get
A = 1000 x 5² = 1000 x 25 = 25000.
Thus 5^{x + 2} = 25000. Ans.

Interchanging L.H.S. and R.H.S. in Law 4, we have
a^{m - n} = a^m⁄aⁿ
Applying this here, we get
B = 5^{x - 2} = 5^x⁄5²
By data 5^x = 1000. Using this here, we get
B = 1000⁄5² = 1000⁄25 = 40.
Thus 5^{x - 2} = 40. Ans.

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Solved Example 4 of Math Exponents

Find x so that (4x)³ = 4³³

Solution to Example 4 of Math Exponents:

Here the exponents are 3 on the L.H.S. and
3³ on the R.H.S.
To find x which is in the base of L.H.S.,
let us make the exponents same.
We know 3³ = 3 x 3 x 3 = 3 x 9 = 9 x 3.
R.H.S. = 4³³ = 4^{9 x 3}
Interchanging L.H.S. and R.H.S. in Law 2, we have
a^mn = (a^m)ⁿ
Applying this here, we get
R.H.S. = 4^{9 x 3} = (4⁹)³
L.H.S. = (4x)³
So, we have
(4x)³ = (4⁹)³
Since the exponents are equal, the bases should be equal.
∴ (4x) = (4⁹)
⇒ x = 4⁹⁄4 = (4⁹)⁄4¹
We know a^m⁄aⁿ = a^{m - n}
Applying this here, we get
x = 4^{9 - 1} = 4⁸. Ans.

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Solved Example 5 of Math Exponents

Solved Example 5 of Math Exponents :

If p⁄q = (2⁄3)³ + (3⁄2)^-3, what is (p⁄q)^-10 ?

Solution to Example 5 of Math Exponents:

We know (a⁄b)^-n = (b⁄a)ⁿ( See Law 3)
Applying this, (3⁄2)^-3 = (2⁄3)³.
By data, p⁄q = (2⁄3)³ + (3⁄2)^-3.
= (2⁄3)³ + (2⁄3)³
= 2{(2⁄3)³}
We know (a⁄b)^m = a^m⁄b^m (See Law 7)
Applying this, (2⁄3)³ = (2³)⁄(3³) = 8⁄27
∴ p⁄q = 2{(2⁄3)³} = 2{8⁄27} = 16⁄27
⇒ (p⁄q)^-10 = (16⁄27)^-10
We know (a⁄b)^-n = (b⁄a)ⁿ.( See Law 3)
Applying this, (16⁄27)^-10 = (27⁄16)¹⁰.
∴ (p⁄q)^-10 = (16⁄27)^-10 = (27⁄16)¹⁰. Ans.

Exercise of Math Exponents :
Application of Laws of Exponents

Problems on Math Exponents :

1. Simplify (i) (4x³)² x (3x³)² x (3xy)⁴ (ii) a^{x + y - z} x a^{y + z - x} x a^{z + x - y}
2. With what should you multiply 2^a to get 2^{a + 2} and 2^{a - 2}
3. Simplify (x^a⁄x^b)^{a + b} x (x^b⁄x^c)^{b + c} x (x^c⁄x^a)^{c + a}

For the Answers, see at the bottom of the page.

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Answers to Exercise of Math Exponents :
Application of Laws of Exponents

Answers to Problems on Math Exponents :

1. 1296x¹⁰y⁴
2. 1⁄4
3. 1.