LAWS OF EXPONENTS

∙ Product Law :

The product of two powers with the same base equals that base raised to the sum of the exponents. 

If x is any nonzero real number and m and n are integers, then

x^m ⋅ xⁿ = x^{m + n}

Example :

3⁴ ⋅ 3⁵ = 3^{4 + 5}

= 3⁹

∙ Quotient Law :

The quotient of two non zero powers with the same base equals the base raised to the difference of the exponents.

If x is any nonzero real number and m and n are integers, then

x^m ÷ xⁿ = x^{m - n}

Example :

3⁷ ÷ 3⁵ = 3^{7 - 5}

= 3²

∙ Power Law :

A power raised to another power equals that base raised to the product of the exponents.

If x is any nonzero real number and m and n are integers, then

(x^m)ⁿ = x^mn

Example :

(3²)⁴ = 3²⁽⁴⁾

= 3⁸

Some other laws of exponents : 

1. A product raised to a power equals the product of each factor raised to that power. 

If x and y are any nonzero real numbers and m is any integer, then

(xy)^m = x^m ⋅ y^m

Example :

(3 ⋅ 5)² = 3² ⋅ 5²

= 9 ⋅ 25

= 225

2. A quotient raised to a positive power equals the quotient of each base raised to that power.

If x and y are any nonzero real numbers and m is a positive integer, then

(x/y)^m = x^m/y^m

Example :

(3/5)² = 3²/5²

3. A quotient raised to a negative power equals the reciprocal of the quotient raised to the opposite (positive) power.

If x and y are any nonzero real numbers and m is a positive integer, then

(x/y)^-m = (y/x)^m

Example :

(5/3)^-2 = (3/5)²

4. If a power is moved from numerator to denominator or denominator to numerator, the sign of the exponent has to be changed.

x^-m = 1/x^m

Example :

3^-2 = 1/3²

5. For any nonzero base, if the exponent is zero, its value is 1.

x⁰ = 1

Example :

3⁰ = 1

6. For any base base, if there is no exponent, the exponent is assumed to be 1.

x¹ = x

Example :

3¹ = 3

7. If an exponent is transferred from one side of the equation to the other side of the equation, reciprocal of the exponent has to be taken. 

x^m/n = y ----> x = y^n/m

Example :

x^2/3 = 3

x =  3^3/2

8. If two powers are equal with the same base, exponents can be equated.

a^x = a^y ----> x = y

Example :

3^m = 3⁵ ----> m = 5

9. If two powers are equal with the same exponent, bases can be equated.

x^a = y^a ----> x = y

Example :

k³ = 5³ ----> k = 5

Important Note

Many students do not know the difference between

(-3)² and -3²

Order of operations (PEMDAS) dictates that parentheses take precedence.

(-3)² = (-3) ⋅ (-3)

(-3)² = 9

Without parentheses, exponents take precedence :

-3² = -3 ⋅ 3

-3² = -9

The negative is not applied until the exponent operation is carried through. We have to make sure that we understand this. So, we will not make this common mistake.

Sometimes, the result turns out to be the same, as in.

(-2)³ and -2³

We have to make sure why they yield the same result.

Solved Problems

Problem 1 :

If a^-1/2 = 5, then find the value of a.

Solution :

a^-1/2 = 5

a = 5^-2/1

a  =  5^-2

a = 1/5²

a = 1/25

Problem 2 :

If n = 1² + 1⁴ + 1⁶ + 1⁸ +...............+ 1⁵⁰, then find the value of n.

Solution :

1 to the power of anything is equal to 1.

1² = 1

1⁴ = 1

1⁶ = 1

List out all the exponents.

2, 4, 6, 8, ........., 48, 50

Divide each element by 2.

1, 2, 3, 4, ........., 24, 25

We can clearly see there are 25 terms in the series shown below.

1² + 1⁴ + 1⁶ + 1⁸ + ......... + 1⁵⁰

Therefore, n is the sum of twenty-five 1's.

n = 1² + 1⁴ + 1⁶ + 1⁸ + ......... + 1⁵⁰

n = 25

Problem 3 :

If 4^{2n + 3} = 8^{n + 5}, then find the value of n.

Solution :

4^{2n + 3} = 8^{n + 5}

(2²)^{2n + 3} = (2³)^{n + 5}

2^{2(2n + 3)} = 2^{3(n + 5)}

Equate the exponents.

2(2n + 3) = 3(n + 5)

4n + 6 = 3n + 15

n = 9

Problem 4 :

If 2^x/2^y = 2³, then find the value x in terms of y.

Solution :

2^x/2^y = 2³

2^{x - y} = 2³

x - y = 3

x = y + 3

Problem 5 :

If a^x = b, b^y = c and  c^z = a, then find the value of xyz.

Solution :

a^x = b

Substitute a = c^z.

(c^z)^x = b

c^zx = b

Substitute c = b^y.

(b^y)^zx = b

b^xyz = b

b^xyz = b¹

xyz = 1

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