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Exponents and Roots

SECTION A Family Letter: Exponents

Dear Family,

The student will be learning about exponents and the properties associated with them. An exponent is a number that represents how many times the base is to be multiplied by itself. A number produced by raising a base to an exponent is called a power.

Using exponents will allow the student to write repeated multiplication in a more efficient way. This is how the student will write and evaluate exponents.

Write in exponential form.

2 • 2 • 2 • 2

2 • 2 • 2 • 2 ’ 24 Identify how many times 2 is a factor.

Simplify 43.

43 ’ 4 • 4 • 4 Find the product of three 4’s.

( 64

Sometimes the exponent will be zero. The zero power of any number except 0 equals 1.

When simplifying expressions with exponents, remind the student to follow the order of operations.

There are important relationships that exist between exponents and the operations of multiplication and division. The student will learn how to multiply and divide powers with the same base by using some of these basic properties of exponents.

Multiplying

When multiplying powers with the same base, keep the base constant and add the exponents.

Multiply. Write the product as one power.

56 • 52

56(2 Add the exponents.

58

Dividing

When dividing powers with the same base, keep the base constant and subtract the exponents.

Divide. Write the quotient as one power.

[pic]

1214–9 Subtract the exponents.

125

Exponents and Roots

SECTION A Family Letter: Exponents continued

When multiplying or dividing powers with the same base it is important to remember not to multiply or divide the bases. This property only applies when bases are the same. Powers with unlike bases cannot be combined using the two properties shown in the examples.

It is possible to raise a power to a power. To simplify a power raised to a power, keep the base and multiply the exponents.

Since it is possible to have a sum or difference that is a negative number, the student will learn how to evaluate expressions with negative exponents. A number raised to a negative exponent equals 1 divided by that number raised to the opposite of the exponent. This is how the student will learn to evaluate expressions with negative exponents.

Divide. Write the quotient as one power.

[pic]

5−2 The bases are the same so subtract the exponents.

[pic] Write the reciprocal and change the sign of the exponent.

[pic] Simplify.

Scientific notation is a shorthand way to write very small or very large numbers. Numbers in scientific notation are written as the product of a decimal and ten with an exponent.

To change numbers from scientific notation to standard notation, move the decimal point the number of places indicated by the exponent, to the right for a positive exponent and to the left for a negative exponent. Similarly, to convert numbers from standard to scientific notation, move the decimal point and multiply by ten to the power of the number of places the decimal point moves.

Multiplying and dividing numbers in scientific notation works the same way as with other exponents.

Have the student explain the purpose of exponents and the many ways they are used in mathematics. Verbalizing this information helps understanding this material.

Sincerely,

Exponents and Roots

SECTION A At-Home Practice: Exponents

Write using exponents.

1. ((4) • ((4) • ((4) 2. 3 • 3 • 3 • 3 • 3 • 3 3. 9 • 9 4. t • t • t

Simplify.

5. 43 6. ((3)4 7. 64 8. ((1)9

9. (2 + 43) 10. (27 – 32) 11. 25 + (8 • 43) 12. (12 − 72)

Multiply. Write the product as one power.

13. 64 • 63 14. 53 • 56 15. 82 • 87 16. k8 • k2

Divide. Write the quotient as one power.

17. [pic] 18. [pic] 19. [pic] 20. [pic]

Simplify the powers of 10.

21. 10−3 22. 10−6 23. 10−8 24. 10−1

Simplify.

25. 3−3 26. [pic] 27. [pic] 28. 74 • 7−5

Write each number in standard or scientific notation.

29. 9,630,000 30. 2.7 ( 10−5 31. 0.0015

Answers: 1. (−4)3 2. 36 3. 92 4. t3 5. 64 6. 81 7. 1296 8. −1 9. 66 10. 18 11. 537 12. −37 13. 67 14. 59

15. 89 16. k10 17. 52 18. r 19. 86 20. p4 21. 0.001 22. 0.000001 23. 0.00000001 24. 0.1 25. [pic] 26. [pic]

27. 16 28. [pic] 29. 9.63 ( 106 30. 0.000027 31. 1.5 ( 10−3

Exponents and Roots

Family Fun: Match It Up!

Directions

Using the numbers in the right-hand column, make the number sentences true in the left-hand column. You may use some numbers more than one time and some numbers not at all. See how well you can match up the correct answers to the number sentences.

|Number Sentences |x equals ______________ |Possible Answers for x |

| | | |

| 1. 3x ’ [pic] | | 0 |

| | |10 |

| 2. 7.2 ( 10x ( 0.072 | |−5 |

| 3. x12 ( 1 | | 2 |

| 4. 8x ( 83 ( 8(2 | | 7 |

| 5. x4 ( 10,000 | | 3 |

| 6. [pic] ( 49 | |−4 |

| | |1 |

| 7. 0.001 ( 10x | | 5 |

| 8. 5x ( 625 | |−3 |

| 9. 12 ( 24 • x(1 | |−2 |

|10. 6x • 6(3 ( 36 | | 6 |

|11. [pic] ( 1 | | 4 |

| | |−1 |

|12. [pic] ( 10x | |−10 |

Answers: 1. −2 2. −2 3. 1 4. −5 5. 10 6. 6 7. −3 8. 4 9. 2 10. 5 11. 3 12.

Exponents and Roots

SECTION B Family Letter: Roots

Dear Family,

The student will be learning about squares and square roots. Although every positive number has two square roots, most of the time you only write the non-negative square root, better known as the principal square root.

The student will also learn to recognize a perfect square and use this knowledge to evaluate expressions and estimate the square roots of numbers that are not perfect squares.

Find the two square roots of 81.

[pic] ’ 9 9 is a solution, since 9 • 9 ’ 81.

−[pic] ’ −9 −9 is also a solution since

−9 • −9 ’ 81.

Remember to follow the order of operations when simplifying expressions with exponents.

Simplify the expression, 4[pic] + 9.

4[pic] + 9

4(8) + 9 Simplify the square root.

32 + 9 Multiply.

41 Add.

You can use the square roots of perfect squares to estimate the square roots of other numbers.

The square root of 55 is between two integers.

Name the integers. Explain your answer.

[pic] Think: What perfect squares

are close to 55?

72 ’ 49 49 < 55

82 ’ 64 64 > 55

7 < [pic] < 8

[pic] is between 7 and 8 because 55 is between 49 and 64.

Some square roots can be written as decimals that terminate or repeat. Other square roots, when written as a decimal, go on forever without repeating or terminating. These numbers are called irrational numbers.

There is one last set of numbers that encompasses all the sets of numbers the student has studied. The set of real numbers includes the set of whole numbers, integers, rational numbers, and irrational numbers. However, there are some

Exponents and Roots

SECTION B Family Letter: Roots continued

numbers that are not real numbers, like the square root of a negative number. The student will learn to identify the set or sets of numbers to which a given number belongs.

The student will also study a special kind of triangle called a right triangle. Right triangles have one right, or 90º, angle. The side opposite the right angle, called the hypotenuse, is the longest side. The shorter sides, that form the right angle, are called legs.

The student will learn an important formula called the Pythagorean Theorem. This theorem relates the length of the legs of a right triangle to the length of the hypotenuse.

[pic]

The Pythagorean Theorem states that the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. Sometimes, the formula is a little easier to understand.

a2 + b2 ’ c2

( ( (

legs hypotenuse

Find the length of the hypotenuse.

a2 + b2 ’ c2 Pythagorean Theorem

52 + 122 ’ c2 Substitute values.

25 + 144 ’ c2 Simplify powers.

169 ’ c2

[pic] ’ [pic] Find the square root.

13 ’ c

The Pythagorean Theorem has many useful applications. One application that the student will learn is finding distance on the coordinate plane. The distance between two points (x1, y1) and (x2, y2) is [pic]. This is called the Distance Formula, and it derives directly from the Pythagorean Theorem.

Powers, roots, real numbers, and the Pythagorean Theorem are all ideas at the foundation of the student math education. Understanding them is essential to his or her progress.

Sincerely,

Exponents and Roots

SECTION B At-Home Practice: Roots

Find the two square roots of each number.

1. 36 2. 121 3. 64 4. 900

Simplify each expression.

5. [pic] − 4 6. [pic] − 9 7. [pic] + 5

Each square root is between two integers. Name the integers.

8. [pic] 9. −[pic] 10. [pic]

Use a calculator to find each value. Round to the nearest tenth.

11. [pic] 12. [pic] 13. [pic]

Find a real number between each pair of numbers.

14. 4[pic] and 4[pic] 15. [pic] and 5 16. [pic] and 0

Find the length of the hypotenuse in each triangle.

17. 18.

Find the distance between each pair of points. Round to the nearest tenth.

19. (2, 3) and (4, 1) 20. (−4, 2) and (3, (1)

Answers: 1. 6, −6 2. 11, −11 3. 8, −8 4. 30, −30 5. 1 6. 0 7. 12 8. 3 and 4 9. −9 and −8 10. 8 and 7 11. 8.5

12. 5.9 13. 28.7 14. 4[pic] 15. 4 16. [pic] 17. c ’ 34 18. c ’ 19.8 19. 2[pic] 20. 7.6

Exponents and Roots

Family Fun: Be a Square

Directions

Cut out the cards and shuffle them. Randomly place the cards face down. Taking turns with your partner, choose two cards and try and match the square root to the perfect square. When you find a pair, keep the cards. If you do not have a match, return each card face down in its original spot. The player with the most matches wins.

|[pic] |[pic] |[pic] |[pic] |

|[pic] |[pic] |[pic] |[pic] |

|2 |3 |4 |5 |

|6 |7 |8 |9 |

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Vocabulary

These are the math words we are learning:

scientific notation

a shorthand way of writing very large or very small numbers using powers of 10

Vocabulary

These are the math words we are learning:

Density Property

a property that states that between any two real numbers there is another real number. This property is also true for rational number–?°?”‚ ‚ƒÚƒæƒ4„•„›„Ì„[…a…•…¢…¦…ñ…@†H†J†L†N†‚ˆ®ˆÞˆôˆýûùýs, but not for whole numbers and integers.

hypotenuse the side on a right triangle across from the right angle

irrational number number written as a decimal that is not terminating or repeating

leg one of the two sides that make up the right angle in a right triangle

perfect square a number that has integers as its square root

principal square root

the non-negative square root that appears on the calculator

Pythagorean Theorem in any right triangle, the sum of the squares of the legs is equal to the square of the length of the hypotenuse

real numbers the set of numbers that consists of the set of rational numbers and the set of irrational numbers

square root a number that when multiplied by itself to form a product is the square root of that product.

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