EXPONENTS AND RADICALS

Exponents

Exponent is used to indicate repeated multiplication of the same number.

An expression like 5⁴ is called a power. The number 5 is the base and 4 is the exponent.

5⁴ = 5 ⋅ 5 ⋅ 5 ⋅ 5

The exponent 4, indicates the number of times the base appears as a factor. An exponent is also called a power

In evaluating the power of 5⁴, we have four factors of 5.

5⁴ is read "5 to the fourth power".

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

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.

Solving Problems on Exponents

Problem 1 :

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 2 :

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 3 :

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

Problem 4 :

If k(2^x) = 2^x+3 - 2^x, then solve for k.

Solution :

Using laws of exponents, we have

k(2^x) = 2^x ⋅ 2³ - 2^x

k(2^x) = 2^x ⋅ 8 - 2^x

k(2^x) = 2^x(8 - 1)

k(2^x) = 2^x(7)

Divide each side by 2^x.

k = 7

Radicals

"Radicals" is a common concept in algebra. In fact, we think of radicals as reversing the operation of an exponent.

Hence, instead of the “square” of a number, we “square root” a number; instead of the “cube” of a number, we “cube root” a number to reverse the square to find the base.

Square roots are the most common type of radical used in algebra.

If 'a' is a positive real number, then the principal square root of a number 'a' is defined as

√a = b if and only if a = b²

If something is given like ³√a, then 3 is called the root or index; hence, ³√a is called the cube root or third root of 'a'.

In general,

ⁿ√a = b if and only if a = bⁿ

If n is even, then 'a' and 'b' must be greater than or equal to zero. If n is even, then 'a' and 'b' must be any real number.

ⁿ√a can be written as a^1/n.

or

ⁿ√a = a^1/n

Laws of Radicals

Law 1 :

ⁿ√(ab) = ⁿ√aⁿ√b

Law 2 :

ⁿ√(a/b) = ⁿ√a/ⁿ√b

Law 3 :

ⁿ√(aⁿ) = a

Law 4 :

ⁿ√(a^m) = a^m/n

Note :

ⁿ√a + ⁿ√b ≠ ⁿ√(a + b)

ⁿ√a - ⁿ√b ≠ ⁿ√(a - b)

Solving Problems on Radicals

Problem 5 :

Evaluate :

√64 + √196

Solution :

Because 64 and 196 are perfect squares, we can find the square root of 64 and 194 as shown below.

√64 + √196 = 8 + 14

= 22

Problem 6 : 

(√3)³ + √27

Answer :

(√3)³ + √27 = (√3 ⋅ √3 ⋅ √3) + √(3 ⋅ 3 ⋅ 3)

= (3 ⋅ √3) + 3√3

= 3√3 + 3√3

Problem 7 : 

Solve for x :

1/³√x = 2

Solution :

1/³√x = 2

1/x^1/3 = 2

x^-1/3 = 2

x = 2^-3

x = 1/2³

x = 1/8

Problem 8 : 

Simplify :

³√(x⁶y⁹)

Solution :

= ³√(x⁶y⁹)

= (x⁶y⁹)^1/3

= (x⁶)^1/3(y⁹)^1/3

= x^6/3y^9/3

= x²y³

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