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Home , Square root, Subtraction

EXPONENTS AND RADICAL REVIEW Prep for Chapter 10 Exponents and Radicals Review I. Inverse Operations Review INVERSE Subtraction Subtraction Addition Division Multiplication Root Squaring Squaring Square Root

Finding a root of a is the inverse operation of raising a number to a power. (You can use other roots and powers besides 2). Exponents and Radicals Review

II. Exponents The exponential form of a number is below. The exponent tells us how many times the will be multiplied by itself. The base is the number being multiplied. A number produced by raising a base to an exponent is called a power. 27 and 33 are equivalent. Base Exponent 2 7 Exponents and Radicals Review

Examples: Write in exponential form.

A. 4 • 4 • 4 • 4• 4 Identify how many times 4 5 4 • 4 • 4 • 4 = 4 is a factor.

B. (–6) • (–6) • (–6) Identify how many times –6 (–6) • (–6) • (–6) = (–6)3 is a factor. Reading Math Read (–6)3 as “–6 to the 3rd power" or "–6 cubed”. Exponents and Radicals Review

III. Radical Expressions

A. Terminology and Symbols

This symbol is the radical or the radical

index radical sign n a radicand

The expression under the radical sign is the radicand. The index defines the root to be taken. Exponents and Radicals Review IV. Square Roots A square root of any positive number has two roots – one is positive and the other is negative. For example, the square root of 64 is both 8 and -8. When simplifying square roots (as opposed to solving them), we only use the positive, or principal, root.  − The symbol The square root The above symbol represents the of a negative represents the negative root of a number is a non- positive or principal number. . root of a number. Ex: - 64 simplified Why? Ex: 64 simplified is -8 is 8 Exponents and Radicals Review

To summarize, A negative outside the radical of a square root means you take the negative of the root. A negative inside the radical of a square root means it is a non-real number. Examples: 6 100  10 36 0.81  0.9

9 non-real # Exponents andRdicals Radicals Review

V. Cube Roots 3 a A cubed root is asking “What base is multiplied three times to get the radicand?” In other words, “What times itself, times itself again, give you the radicand? Examples:

3 3 27  3 8 2

A of any positive number is positive. A cube root of any is negative. Exponents and Radicals Review th VI. n Roots An of any number a is a number whose nth power is a. Examples:

34  81 4 81  3

24  16 4 16  2

5 2 32 5 32 2 Exponents and Radicals Review th n Roots Be careful with negatives inside the radical. Examples: Do you see a pattern? 5 1 1 If there is a negative under the radical and 4 16 Non-real number the index is positive, it is a non-real number.

6 If there is a negative 1 Non-real number under the radical and the index is negative, 3 27 3 the root will be negative. VII. PRIME FACTORIZATION Breaking down a number into its prime factorization will help you simplify radical expressions in 10-1. A. PRIME AND COMPOSITE A whole number, greater than 1, for which the only factors are 1 and itself is called a .

Examples: 2, 3, 5, 7, 11, 13, 17, 19

A whole number, greater than 1, that has more than two factors is called a composite number. Examples: 4, 6, 8, 9, 10, 12, 14, 15, 20

0 and 1 are NEITHER prime nor composite. B. Finding the prime factorization of a number A whole number expressed as the of prime numbers is called the prime factorization of the number.

Ex. 1: Find the prime factorization of 90 Make a factor tree 90 What are some factors of 90? 9 10 Can we break down 9 and 10 further? Write factors for 3 3 5 2 each number that is composite. 90 = 2 ∙ 3 ∙ 3 ∙ 5 표푟 90 = 2 ∙ 32 ∙ 5 Circle the prime numbers. Notice how the numbers are in order from least to greatest.