Exponents and Radicals



Unit 1 Laws of Exponents

An exponent is a numeric or algebraic expression that indicates how many times a quantity is to be multiplied by itself.

In an algebraic or numerical expression, the base is the number or variable with an exponent. The base can be a natural number N an integer Z, a fraction or a decimal Q.

Laws of signs in multiplication: + x + = + − x − = + + x − = − − x + = −

• Any power with a positive base is positive.

• Any power with a negative base is positive if the exponent is an even number.

• Any power with a negative base is negative if the exponent is an odd number.

Laws of Exponents :

1. When two or more powers with like bases are multiplied, the product has the same base and the exponents are added.

a m × a n = a m + n

2. When two powers with like bases are divided, the quotient has the same base and the exponents are subtracted.

a m ÷ a n = a m − n , a ≠ 0

3. When the base has a negative exponent, the base is inverted and the expression becomes positive.

[pic] =[pic] , where m Є Q +

4. Any expression other than 0 that has the exponent 0 is equal to 1.

a 0 = 1

5. To raise a power m with base a to a power n, multiply the exponents.

(am) n = a m × n

6. To raise a product in exponential form to a power n , multiply the exponents of each of the factors

by n.

(a b c) m = a m b m c m where a , b , and c represent exponential expressions.

7. To raise a quotient of two expressions to a power n , multiply the exponents of each expression by n.

[pic] or (a ÷ b) m = a m ÷ b m , where b ≠ 0

Rules for Applying the Laws of Exponents:

1. The power of a positive base is positive.

2. The power of a negative base is positive if the exponent is an even number.

3. The power of a negative base is negative if the exponent is an odd number.

4. When the base has a negative exponent, invert the base and change the sign of the exponent to obtain an equivalent expression in which the exponent is positive.

5. When an expression preceded by a negative sign has an exponent, each term must be raised to this power, including the digit −1.

Scientific Notation

When we multiply a decimal number by a power of 10 (i.e. 10, 100, 1000, 10 000, etc) the decimal point is moved to the right by as many places as there are zeros in the power of 10.

When we divide a decimal number by a power of 10 the decimal point is moved to the left by as many places as there are zeros in the power of 10.

In scientific notation, a number is always expressed in the form a × 10 n, where n is an integer and a is a decimal, such that 1 ≤ a < 10.

The number a × 10 n can be converted to a decimal by moving the decimal point in a to the right by n places , where n is a natural number and 1 ≤ a < 10

The number a × 10 − n can be converted to a decimal by moving the decimal point in a to the left by n places , where n is a natural number and 1 ≤ a < 10

Procedure to express a number in scientific notation :

1. Place the decimal to the right of the first non-zero digit.

2. Count the number of places the decimal point was moved.

3. Write this number as the base 10 exponent :

a) this exponent is positive if the decimal point was moved to the left.

b) this exponent is negative if the decimal point was moved to the left.

Note : To assign an exponent to several variables or numbers , place these between parentheses. Otherwise , the exponent will apply only to the closest variable or number.

Unit 2 Simplifying Algebraic or Numerical Expressions Written in Exponential Form

Law of priority of operations: BEDMAS

1. Perform operations between parentheses inner brackets to outer bracketsfirst.

2. Perform the Laws of Exponents on any exponential terms with the same base.

3. Next perform multiplication and / or division operations.

4. Lastly perform addition and subtraction of like terms.

Add similar monomials (like terms).

Note:

• Two monomials are similar if they are composed of the same variables with the same exponents

• The number 1 raised to any exponent is equal to 1.

• The power of a negative base is positive if the exponent is even numbered.

To simplify an algebraic or a numerical expression whose terms are in exponential form:

1. Observe the rule of priority of operations.

2. Apply the appropriate laws of exponents.

3. Assign the appropriate sign to each power depending on whether the exponent is

even − or odd − numbered.

4. If necessary, convert all negative exponents to positive exponents.

Unit 3 Converting an Expression, Containing a Radical, to Exponential Form and Vice Versa

Radicals: [pic] [pic]

[pic]= the radicand; √ = the radical sign; n √ n = the root index

[pic] [pic]

Place under the radical sign only those radicands that appear under radicals with the same root index.

Conjugates: Binomials containing radicals consisting of identical terms , but linked together by an opposite sign are called conjugates and their product is always a rational number.

[pic]

A numerical coefficient is a number that multiplies a variable or an arithmetic expression.

• The nth root of a number x is a quantity whose nth power is equal to x.

• The nth power of a number is the product of n factors equal to that number.

• The radical sign ( √ ) represents the operation which consists of extracting the root of a number.

• The radicand is the expression under the radical sign.

• When the root index = 2 , it is understood.

• When the exponent of a number = 1 , it is understood.

• If a = b and b = c , then a = c.

• (a n ) m = a n × m

To convert to exponential form a radical that is the power of a single base :

1. Remove the radicand from the radical and use it as the base of the exponential expression.

2. Divide the exponent of the radicand by the root index and make the resulting fraction the exponent of this base.

3. Convert to the smallest base if the base is a number and , if necessary , simplify the resulting expression by applying the fifth law of exponents.

To convert an exponential expression to a radical :

1. Place the base of the exponential expression and the numerator of the exponent under a radical sign.

2. Make the denominator of the exponent the root index.

3. Calculate the radicand , if necessary.

[pic] [pic]

To convert a numerical or an algebraic expression of the form a m × n √ (a p) to an exponential expression in simplest form :

1. Convert the term under the radical sign to exponential form.

2. Convert the bases to the smallest possible , if necessary.

3. Apply the appropriate laws of exponents.

Unit 4 The Sum , Difference , Product and Quotient of Numerical Expressions Containing Square Roots

To break a number down into factors is to find all the numbers whose product is equal to this number.

Two or more radicals are similar if they have identical root indexes and radicands , regardless of the value of the numerical coefficient.

The expression 3 √2 means 3 × √2 . If you need to multiply two expressions containing radicals , the commutative and associative properties allow you to group the terms of the multiplication without changing the value of the result.

To calculate a product involving radicals:

1. Multiply the coefficients of the radicals.

2. Place the radicands with the same root index under the same radical sign and keep the multiplication sign between them.

3. Simplify the number under the radical and multiply the extracted quantity by the numerical coefficient.

[pic] [pic]

Note : When multiplying radicals , place under the same radical sign only those radicands that appear under radicals with the same root index.

Note that in mathematics, result must never have an irrational denominator. The denominator of the resulting expression must be rationalized as follows:

1. Multiply the numerator and the denominator of the expression to be rationalized by the radical in the denominator of this expression.

2. Extract the root of the denominator and multiply it by the numerical coefficient of the radical, as applicable.

Unit 5 Operations on Polynomials containing Square Roots

The distributive property applies to multiplication and to addition or subtraction. This makes it possible to multiply the number or variable in front of the parentheses by each of the terms inside the parentheses, without changing the value of the expression.

A polynomial is a form of algebraic expression composed of several terms linked by addition and / or subtraction signs. More particularly, a binomial is a polynomial with two terms , and a trinomial is a polynomial with three terms.

Binomials containing radicals consisting of identical terms , but linked together by an opposite sign are called conjugates and their product is always a rational number.

To calculate the product of polynomials containing radicals with the same root index:

1. Apply the distributive property of multiplication over addition or subtraction as many times as necessary.

2. Perform the multiplications one by one.

3. Reduce the radicals , if necessary.

4. Add or subtract similar terms.

To rationalize a denominator consisting of a binomial containing one or two radicals:

1. Multiply the numerator and the denominator of the expression to be rationalized by the conjugate of its denominator.

2. Find the products in the denominator and , if applicable , in the numerator.

3. Simplify, if necessary,

4. Simplify the radicals in the numerator and find the common factor, if necessary.

5. Simplify and make the denominator positive, if necessary.

To rationalize a denominator composed of a monomial containing a radical, when the numerator consists of a binomial containing a radical:

1. Multiply the numerator and the denominator by the radical in the denominator.

2. Find the products.

3. Simplify the radicals, if necessary,

4. Factor out the common factor in the numerator and simplify, if necessary.

Exponents and Radicals Essential Formulae

Laws of Exponents :

1. a m × a n = a m + n

2. a m ÷ a n = a m − n , a ≠ 0

3. a − m = [pic] , where m Є Q +

4. a 0 = 1

5. (a m) n = a m × n

6. (a b c) m = a m b m c m , where a , b , and c represent exponential expressions.

7. [pic]= [pic] or (a ÷ b) m = a m ÷ b m , where b ≠ 0

The result of simplifying an exponential expression should not have a negative exponent.

To assign an exponent to several variables or numbers , place these between parentheses. Otherwise , the exponent will apply only to the closest variable or number.

Remember the Law of priority of operations and to add similar monomials (like terms).

Scientific Notation : A number is always expressed in the form a × 10 n , where n is an integer and a is a decimal , such that 1 ≤ a < 10.

The number a × 10 n can be converted to a decimal by moving the decimal point in a to the right by n places , where n is a natural number and 1 ≤ a < 10

The number a × 10 − n can be converted to a decimal by moving the decimal point in a to the left by n places , where n is a natural number and 1 ≤ a < 10

Radicals : [pic] [pic] [pic]

[pic]= the radicand ; √ = the radical sign ; n √ n = the root index

[pic] [pic]

The result of simplifying an radical expression should not have a radical in the denominator

.

Conjugates : Binomials containing radicals consisting of identical terms , but linked together by an opposite sign are called conjugates and their product is always a rational number.

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