The inverse operation to squaring a number is to find its square roots. b is a square rootof a if b2 = a. • All positive real numbers have two real-valued square roots: one positive and one negative. • Zero has only one square root, itself. • For any negative real number there are no real-valued square roots.
Let a represent a positive real number. Then • is the positive square root of a. The positive square root is also called the principle square root. • is the negative square root of a. The symbol used to denote the positive square root of a real number is called a radical sign .
The numbers 25, , and 0.36 are perfect squares because their square roots are rational numbers. Radicals that cannot be simplified to rational numbers are irrational numbers. The radical notation represents the exact value, but an approximate value can be found using a calculator.
nth Roots b is an nth root of a if bn = a. In the expression , n is called the indexof the radical and a is called the radicand. The expression denotes the principal nth root of a.
Definition of • If n is a positive even integer and a > 0, then is the principal (positive) nth root of a. • If n > 1 is an odd integer, then is the nth root of a. • If n > 1 is an integer, then
A radical with an index of 3 is called a cube root, . • It is helpful to know the following powers in order to simplify radicals.
Definition of • If n is a positive odd integer, then • If n is a positive even integer, then If the variable a is assumed to be nonnegative, then absolute value bars may be dropped.
Pythagorean Theorem a2 + b2 = c2
Radical Functions If n > 1, then f(x) = is a radical function. If n is an even integer, the domain is restricted to nonnegative real numbers. If n is an odd integer, the domain is all real numbers. Example: The domain of is all values where 2t – 6 is greater than or equal to zero, or [3, ∞).
Exercise 1 Simplify the root, if possible. A) 5 B) 4 C) –4 D) Not a real number
Exercise 2 Simplify the expression. Assume that all variables represent positive real numbers. A) 4a6b2 B) 61a15b3 C) –4a6b2 D) 2a3b3
Exercise 3 Find the domain of the function. A) [4, ∞) B) (4, ∞) C) All real numbers D) Ø (The empty set)
Let a be a real number, and let n be an integer such that n > 1. If is a real number, then Moreover, if m is a positive integer such that m and n share no common factors, then and
Properties of Exponents Let a and b be nonzero whole numbers. Let m and n be rational numbers such that am, an,and bn are real numbers. Description Property • Multiplying like bases aman = am + n • Dividing like bases • The power rule (am)n = amn • Power of a product (ab)m = ambm • Power of a quotient
Exponent Definitions Let a be a nonzero real number. Let m be a rational number such that amis a real number. Description Definition • Negative exponents • Zero exponent a0 = 1
Exercise 4 Evaluate the expression. 642/3 A) • 16 • –16 D) 19
Exercise 5 Convert the expression to an equivalent expression using rational exponents. Assume that all variables represent positive real numbers. A) 2x–7 B) 2x1/7 C) 2x–1/7 D) (2x)–1/7
Exercise 6 Use properties of exponents to simplify the expression. Assume that all variables represent positive real numbers. A) t6 B) t–6 C) t10 D) t–2
Multiplication and Division Properties of Radicals Let a and b represent real numbers such that and are both real. Then • Multiplication property of radicals • Division property of radicals
The multiplication property of radicals allows us to simplify a product of factors within a radical. However, this rule does not apply to terms that are added or subtracted within the radical. For example:cannot be simplified.
Simplified Form of a Radical Consider any radical expression where the radicand is written as a product of prime factors. The expression is in simplified form if all the following conditions are met. • The radicand has no factor raised to a power greater than or equal to the index. • The radicand does not contain a fraction. • There are no radicals in the denominator of a fraction.
Examples Simplify each expression. Apply the multiplication property of radicals. Apply the division property of radicals.
Exercise 7 Use the multiplication property of radicals to multiply, then simplify the result. A) (14y – 14)9 B) (14y – 14)3 C) D)
Exercise 8 Use the division property of radicals to divide, then simplify the result. Assume that all variables represent positive real numbers. A) 3 B) 3z C) 9 D)
Exercise 9 Simplify the radical. Assume that all variables represent positive real numbers. A) B) C) D)
Two radical terms are said to be like radicals if they have the same index and the same radicand. like radicalsnot like radicals
The process of adding like radicals with the distributive property is similar to adding like terms. The end result is that the numerical coefficients are added and the radical factor is unchanged. Be careful: In general:
Exercise 10 Which two radicals are like? A) B) C) D)
Exercise 11 Add, if possible: A) 28 B) 70 D) Cannot be simplified
Exercise 12 Subtract, if possible: A) B) C) D) Cannot be simplified
The Multiplication Property of Radicals Let a and b represent real numbers such that and are both real. Then To multiply two radical expressions, use the multiplication property of radicals along with the commutative and associative properties of multiplication.
ExampleMultiply the expressions and simplify the result. Commutative and associative properties of multiplication Multiplication property of radicals
When multiplying radical expressions with more than one term, use the distributive property. Example
If is a real number, then . • Patterns from multiplying binomials apply to multiplying radical expressions:
If two radicals have different indices, use the properties of rational exponents to obtain a common index. Rewrite with rational exponents. Bases are equal; add exponents. Find a common denominator. Simplify. Rewrite the expression as a radical.
Exercise 13 Multiply. A) B) –5y + 10 C) D)
Exercise 14 Multiply. A) 70 B) –40 C) 92 D)
Exercise 15 Multiply. A) B) z C) D)
For a radical expression to be in simplified form, no radicals may be in the denominator of a fraction. Not in simplified form: The process to remove a radical from the denominator is called rationalizing the denominator.
Rationalizing the Denominator: One Term The nth root of a perfect nth power simplifies completely. To rationalize a radical expression, use the multiplication property of radicals to create an nth root of an nth power. Create a fifth root of a fifth power in the denominator. Multiply the radicals. Simplify.
Rationalizing the Denominator: Two Terms To rationalize a two-term denominator, multiply the numerator and denominator by the conjugate of the denominator. Multiply numerator and denominator by the conjugate of the denominator, then apply the formula (a + b)(a – b) = a2 – b2.