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## Texas Go Math Grade 8 Unit 1 Study Guide Review Answer Key

**Texas Go Math Grade 8 Unit 1 Exercises** **Answer Key**

**Find the two square roots of each number. If the number is not a perfect square, approximate the values to the nearest 0.05. (Lesson 1.1)**

**Module 1 Real Numbers**

Question 1.

16 ___________

Answer:

\(\sqrt{16}\) = 4, since 4 . 4 = 16

–\(\sqrt{16}\) = -4, since (-4) . (-4) = 16

Two square roots of 16 are +4 and -4.

Question 2.

\(\frac{4}{25}\) _______________

Answer:

Two square roots of \(\frac{4}{25}\) are +\(\frac{2}{5}\) and –\(\frac{2}{5}\)

Question 3.

225 _________

Answer:

\(\sqrt{225}\) = 15, since 15 . 15 = 225

–\(\sqrt{225}\) = -15, since (-15) . (-15) = 225

Two square roots of 225 are +15 and -15.

Question 4.

\(\frac{1}{49}\) _______________

Answer:

Two square roots of \(\frac{1}{49}\) are +\(\frac{1}{7}\) and –\(\frac{1}{7}\)

Question 5.

\(\sqrt{10}\) _______

Answer:

Since 10 is not a perfect square, approximate 10 with two closest perfect squares:

9 < 10 < 1.6

Take square root to find in between what numbers lies \(\sqrt{10}\):

\(\sqrt{9}\) < \(\sqrt{10}\) < \(\sqrt{16}\)

Solve:

3 < \(\sqrt{10}\) < 4

We now know that \(\sqrt{10}\) lies between 3 and 4 but since 10 is closer to 9, \(\sqrt{10}\) is closer to 3. To find a better approximation pick a few number between 3 and 4 but closer to 3 and square them to see where they lie relating to 10:

3.10^{2} = 9.61 < 10

3.15^{2} ≈ 9.92 < 10

3.2^{2} = 10.24 > 10

We see that 3.15^{2} is closer to 10 than 3.2^{2} so we can conclude that the best approximation to the nearest 0.05 is 3.15

Question 6.

\(\sqrt{18}\) _______

Answer:

Since 18 is not a perfect square, approximate 18 with two closest perfect squares:

16 < 18 < 25

Take square root to find in between what numbers lies \(\sqrt{18}\):

\(\sqrt{16}\) < \(\sqrt{18}\) < \(\sqrt{25}\)

Solve:

4 < \(\sqrt{18}\) < 5

We now know that \(\sqrt{18}\) lies between 4 and 5 but since 18 is closer to 16, \(\sqrt{18}\) is closer to 4. To find a better approximation pick a few number between 4 and 5 but closer to 4 and square them to see where they lie relating to 18:

4.15^{2} ≈ 17.22 < 18

4.2^{2} = 17.64 < 18 4.25 ≈ 18.06 > 18

We see that 4.25^{2} is closer to 18 than 4.2^{2} so we can conclude that the best approximation to the nearest 0.05 is 4.25

**Write all names that apply to each number. (Lesson 1.2)**

Question 7.

\(\frac{2}{3}\) _______________

Answer:

Since \(\frac{2}{3}\) isn’t an integer or a whole number but is expressed as a ratio of two integers we can conclude that it is a rational, real number.

Rational, real.

Question 8.

–\(\sqrt{100}\) _______

Answer:

–\(\sqrt{100}\)

-10

Rational, integer, real

Question 9.

\(\frac{15}{5}\) _______________

Answer:

Since \(\frac{15}{5}\) is a ratio of two integers it is a rational real number. But, since

\(\frac{15}{5}\) = 3

it is also an integer and a whole number.

Whole, integer, rational, real

Question 10.

\(\sqrt{21}\) _______

Answer:

\(\sqrt{21}\)

Irrational, real

**Compare. Write <, >, or . (Lesson 1.3)**

Question 11.

Answer:

\(\sqrt{7}\) + 5 ? 7 + \(\sqrt{5}\) Given

\(\sqrt{7}\) is between 2 and 3 Estimate the value of \(\sqrt{7}\)

\(\sqrt{5}\) is between 2 and 3 Estimate the value of \(\sqrt{5}\)

\(\sqrt{7}\) + 5 is between 7 and 8 Use approximations to simply the expressions

7 + \(\sqrt{5}\) is between 9 and 10

\(\sqrt{7}\) + 5 < 7+ \(\sqrt{5}\) Compare

Question 12.

Answer:

6 + \(\sqrt{8}\) ? \(\sqrt{6}\) + 8 Given

\(\sqrt{8}\) is between 2 and 3 Estimate the value of \(\sqrt{8}\)

\(\sqrt{6}\) is between 2 and 3 Estimate the value of \(\sqrt{6}\)

6 + \(\sqrt{8}\) is between 8 and 9 Use approximations to simply the expressions

\(\sqrt{6}\) + 8 is between 10 and 11

6 + \(\sqrt{8}\) < \(\sqrt{6}\) + 8 Compare

Question 13.

Answer:

\(\sqrt{4}\) – 2 ? 4 – \(\sqrt{2}\) Given

\(\sqrt{4}\) = 2 Estimate the value of \(\sqrt{4}\)

\(\sqrt{2}\) is between 1 and 2 Estimate the value of \(\sqrt{2}\)

\(\sqrt{4}\) – 2 = 2 – 2 = 0 Use approximations to simply the expressions

4 – \(\sqrt{2}\) is between 3 and 2

\(\sqrt{4}\) – 2 < 4 – \(\sqrt{2}\) Compare

**Order the numbers from least to greatest. (Lesson 1.3)**

Question 14.

\(\sqrt{81}\), \(\frac{72}{7}\), 8.9

Answer:

First, express au. numbers in (simplified) decimal form so we could easily compare them.

\(\sqrt{81}\) = 9 Simplify (81 is a perfect square)

\(\frac{72}{7}\) = 10.29 Using long division divide 72 by 7

8.9 Already in the simplest form

Now we can compare:

8.9 < 9 < 10.26 (1)

From (1) it now follows that:

From least to greatest: 8.9 < \(\sqrt{81}\) < \(\frac{72}{7}\)

Question 15.

\(\sqrt{7}\), 2.55, \(\frac{7}{3}\)

Answer:

\(\sqrt{7}\) lies between 2 and 3 Estimate the value of \(\sqrt{7}\)

Since $7$ is approximately midway

between $4$ and $9$, hence

2.65^{2} = 6.50

2.6^{2} = 6.76

2.65^{2} = 7.02

Since $2.65^{2} = 7.02$ $\sqrt{7}

\approx 2.65$

2.55 Given

\(\frac{7}{3}\) = 2.33 Divide 7 by 3

Graph on the number line

\(\frac{7}{3}\), 2.55, \(\sqrt{7}\)

**Module 2 Scientific Notation**

**Essential Question**

How can you use scientific notation to solve real-world problems?

Answer:

Scientific notation is used to write very large or very small numbers using fewer digits. See how scientists use this notation to describe astronomical distances, such as the distance between planets, or microscopic distances, such as the length of a blood cell

**Exercises**

Write each number in scientific notation. (Lessons 2.1, 2.2)

Question 1.

3000 _________

Answer:

3000 in the scientific notation is 3 × 10³

Here move the decimal 3 times to the left then it is 3. But the three are less than 10.

Moved the decimal to the left is the exponent and the exponent is positive.

Question 2.

0.000015 _____

Answer:

0.000015 in the scientific notation is 1.5 × 10^{-5}.

Here move the decimal 5 times to the left then it is 1.5. But the three are less than 10.

Moved the decimal to the left is the exponent and the exponent is negative.

Question 3.

25,500,000 _____

Answer:

25,500,000 Given

2.55 Move the decimal point 7 places to the left. Remove the extra zeros.

10, 000, 000 Divide the original number by the result from above.

10^{7} Write the answer as power of 10.

2.55 × 10^{7} Write the product of the resuLts from step 1 and 2

2.55 × 10^{7}

Question 4.

0.00734 _________

Answer:

0.00734 Given

7.34 Place the decimal point

3 Count the number of places the decimal point is moved.

7.34 × 10^{-3} Multiply 7.34 times a power of 10. Since 0.00734 is less than 1, the decimal point moves to the right and the exponent on 10 is negative.

7.34 × 10^{-3}

**Write each number in standard notation. (Lessons 2.1, 2.2)**

Question 5.

5.23 × 10^{4} __________________________

Answer:

5.23 × 10^{4} Given

4 Use the exponent of the power of 10 to determine the number paces to move the decimal point

places

52,300 Place the decimal point Since you are going to write a number greater than 5.23, move the decimal point to the right. Add placeholder zeros if necessary.

52,300

Question 6.

1.05 × 10^{6} __________________________

Answer:

Given that 1.05 × 10⁶

10⁶ = 1000000

1.05 × 1000000 = 1050000.

Question 7.

4.7 × 10^{-1} __________________________

Answer:

Given that 4.7 × 10^{-1}.

10^{-1} = 0.1

4.7 × 0.1 = 0.47.

Question 8.

1.33 × 10^{-5} __________________________

Answer:

1.33 × 10^{-5} Given

5 Use exponent of the power of 10 to determine the number paces to move the decimal point

places

0.0000133 Place the decimal point Since you are going to write a number greater than 1.33, move decimal point to the right. Add placeholder zeros if necessary.

0.0000133

Use the information in the table to write each weight in scientific notation. (Lessons 2.1, 2.2)

Question 9.

Ant _________

Answer:

Given that the weight of the ant = 0.000000661.

0.000000661 in the scientific notation is 6.61 × 10^{-7}.

Here move the decimal 7 times to the left then it is 6.61. But the three are less than 10.

Moved the decimal to the left is the exponent and the exponent is negative.

Question 10.

butterfly _________

Answer:

Given that the weight of the butterfly = 0.00000625.

0.00000625 in the scientific notation is 6.25 × 10^{-6}.

Here move the decimal 6 times to the left then it is 6.25. But the three are less than 10.

Moved the decimal to the left is the exponent and the exponent is negative.

Question 11.

elephant _________

Answer:

Given that the weight of the elephant = 9900

9900 in the scientific notation is 9.9 × 10³

Here move the decimal 3 times to the left then it is 9.9.But the three are less than 10.

Moved the decimal to the left is the exponent and the exponent is positive.

**Texas Go Math Grade 8 Unit 1 Performance Tasks Answer Key**

Question 1.

**Careers In Math Astronomer** An astronomer is studying Proxima Centauri, which is the closest star to our Sun. Proxima Centauri is 39,900,000,000,000,000 meters away.

a. Write this distance in scientific notation.

Answer:

39,900,000,000,000,000 Distance in meters

3.99 Move the decimal point 16 places to the left. Remove the extra zeros

10, 000, 000, 000, 000, 000 Divide the original number by the result from Step 1. Write the answer as power of 10.

10^{16}

3.99 × 10^{16} Write the product of the results from the above

b. Light travels at a speed of 3.0 × 10^{8} m/s (meters per second). How can you use this information to calculate the time in seconds it takes for light from Proxima Centauri to reach Earth? How many seconds does it take? Write your answer in scientific notation.

Answer:

speed = 3.0 × 10^{8} speed in meters per second

\(\frac{3.99}{3}\) = 1.33 Find the quotient by dividing the decimals and using the laws of exponents

\(\frac{10^{16}}{10^{8}}\) = 10^{16-8} = 10^{8}

1.33 × 10^{8} Combine the answers to write the distance in scientific notation

It takes $1.33 \times 10^{8}$ seconds

for the light to reach Earth

c. Knowing that 1 year = 3.1536 × 10^{7} seconds, how many years does it take for light to travel from Proxima Centauri to Earth? Write your answer in standard notation. Round your answer to two decimal places.

Answer:

1.33 × 10^{8} seconds From part b

1 yr = 3.1536 × 10^{7}

Find the quotient by dividing the decimals and using the Laws of exponents

0.4217 × 10^{1} combine the answers to write the distance in scientific notation

4.22 Write in standard notation

It takes $4.22$ years for the light to reach Earth

Question 2.

Cory is making a poster of common geometric shapes. He draws a square with a side length of 4^{3} cm, an equilateral triangle with a height of \(\sqrt{200}\) cm, a circle with a circumference of 8π cm, a rectangle with length \(\frac{122}{5}\) cm, and a parallelogram with base 3.14 cm.

a. Which of these numbers are irrational?

Answer:

\(\sqrt{200}\) cm and 8π cm are irrational Irrational numbers are real number that cannot be expressed in the form \(\frac{a}{b}\)

b. Write the numbers in this problem in order from least to greatest. Approximate π as 3.14.

Answer:

4^{3} = 64 Evaluate the power

\(\sqrt{200}\) Estimate the value of \(\sqrt{200}\)

lies between $14$ and $15$ Since

$200$ is closer to $196 = 14^{2}$,

hence

14.1^{2} = 198.81

14.15^{2} = 200.22

14.2^{2} = 201.64

Since $14.15”{2} = 200.22$

$\sqrt{200} \approx 14.15$

8π = 8 * 3.14 = 25.12 π = 3.14

\(\frac{122}{5}\) = 24.4 Divide 122 by 5

3.14 Given

Graph on the number line

3.14, \(\sqrt{200}\), \(\frac{122}{5}\), 8π, 4^{3} From least to greatest

c. Explain why 3.14 is rational, but π is not.

Answer:

3.14 can be expressed in the form \(\frac{a}{b}\) but π cannot be expressed in fraction form. It gives a value which is neither a terminating or repeating decimal.

3.14 is rational but π is not

**Texas Go Math Grade 8 Unit 1 Mixed Review Texas Test Prep Answer Key**

**Selected Response**

Question 1.

A square on a large calendar has an area of 4220 square millimeters. Between which two integers is the length of one side of the square?

(A) between 20 and 21 millimeters

(B) between 64 and 65 millimeters

(C) between 204 and 205 millimeters

(D) between 649 and 650 millimeters

Answer:

(B) between 64 and 65 millimeters

Explanation:

The area of a square is found by multiplying the side of the square by itself. Therefore, to find the side of the

square, we have to take the square root of the area.

Lets denote with A the area of a square of the calendar and with s each side of the square. We have:

A = 4220

A = s • s

s = \(\sqrt{A}\) = \(\sqrt{4220}\)

Following the steps as in “Explore activity 1” on page 9, we can make an estimation for the irrational number:

Therefore, the side of the square is between 64 and 65 millimeters.

Question 2.

Which of the following numbers is rational but not an integer?

(A) -9

(B) -4.3

(C) 0

(D) 3

Answer:

(B) -4.3

Explanation:

-4.3 is not an integer

Integers are whole numbers that can be positive, negative or zero.

Question 3.

Which statement is false?

(A) No integers are irrational numbers.

(B) All whole numbers are integers.

(C) All rational numbers are real numbers.

(D) All integers are whole numbers.

Answer:

(D) All integers are whole numbers.

Explanation:

All integers are whole numbers is false

Whole numbers are non-negative while integers are negative, positive or zeros.

Question 4.

Which set best describes the numbers displayed on a telephone keypad?

(A) whole numbers

(B) rational numbers

(C) real numbers

(D) integers

Answer:

The rational numbers are displayed on the telephone keypad.

Option B is the correct answer.

Question 5.

In 2011, the population of Laos was about 6.586 × 10^{6} people. What is this number written in standard notation?

(A) 6,586 people

(B) 658,600 people

(C) 6,586,000 people

(D) 65,860,000 people

Answer:

(C) 6,586,000 people

Explanation:

6.586 × 10^{6} Population in 2011

6 Use exponent of the power of 10 to determine the number places to move the decimal point

places

6, 586, 000

Place the decimal point Since you are going to write a number greater than 6.586, move decimal point to the right Add placeholder zeros if necessary.

There are 6, 586, 000 people in $2011$

Question 6.

Which of the following is not true?

Answer:

(D) 6 – \(\sqrt{35}\) < 0 Explanation: a. Since both 16 and 4 are perfect squares we can evaluate them easily: and the statement is TRUE. b. We know that π ≈ 3.14 which means that 4π ≈ 4 × 3.14 = 12.56 Obviously, 12.56 > 12

so,

4π > 12

and the statement is TRUE

c. Lets first evaluate \(\sqrt{18}\) and \(\frac{15}{2}\)

Since 18 is not a perfect square find two closest squares that 1 lies in between:

16 < 18 < 25

\(\sqrt{16}\) < \(\sqrt{18}\) < \(\sqrt{25}\)

4 < \(\sqrt{18}\) < 5

Since 18 is closer to 16 a good approximation for \(\sqrt{18}\) is 4 but we are certain that \(\sqrt{18}\) is not larger or equal to 5

To express \(\frac{15}{2}\) as a fraction use long division to divide 15 by 2

\(\frac{15}{2}\) = 7.5

Now we can see that

\(\sqrt{18}\) + 2 < 5 + 2 = 7

\(\frac{15}{2}\) = 7.5

We can conclude that \(\sqrt{18}\) + 2 < \(\frac{15}{2}\)

and the statement is TRUE.

d. Since

6^{2} = 36

36 > 35

by taking square roots from both sides we can see that

\(\sqrt{35}\) < \(\sqrt{36}\) = 6 but now 6 – \(\sqrt{35}\) > 0

It now follows that the statenient is NOT TRUE.

Question 7.

Which number is between \(\sqrt{50}\) and \(\frac{5 \pi}{2}\)?

(A) \(\frac{22}{3}\)

(B) 2\(\sqrt{8}\)

(C) 6

(D) π + 3

Answer:

(A) \(\frac{22}{3}\)

Explanation:

\(\sqrt{50}\) Given

and \(\frac{5 \pi}{2}\)

\(\sqrt{50}\) lies between 7 and 8 Evaluate the value of \(\sqrt{50}\)

7.0^{2} = 49.00

7.05^{2} = 49.70

7.15^{2} = 51.12

Since $7.05^{2} = 49.7$, $\sqrt{50}

\approx 7.05$

\(\frac{5 * 3.14}{2}\) = 7.85 Evaluate \(\frac{5 \pi}{2}\)

The number should be between 7.05 and 7.85

\(\frac{22}{3}\) = 7.33

Question 8.

What number is indicated on the number line?

(A) π + 4

(B) \(\frac{152}{20}\)

(C) \(\sqrt{14}\) + 4

(D) \(7 . \overline{8}\)

Answer:

Given that the number \(\sqrt{14}\) + 4 indicated on the number line.

\(\sqrt{14}\) + 4 = 7.7416.

Option C is the correct answer.

Question 9.

Which of the following is the number 5.03 × 10^{-5} written in standard form?

(A) 503,000

(B) 50,300,000

(C) 0.00503

(D) 0.0000503

Answer:

The standard form of the number 5.03 × 10^{-5}

5.03 × 0.00001 = 0.0000503

Here move the decimal 5 times to the left then it is 5.03. But the three are less than 10.

Moved the decimal to the left is the exponent and the exponent is negative.

Option D is the correct answer.

Question 10.

In a recent year, about 20,700,000 passengers traveled by train in the United States. What is this number written in scientific notation?

(A) 2.07 × 10^{1} passengers

(B) 2.07 × 10^{4} passengers

(C) 2.07 × 10^{7} passengers

(D) 2.07 × 10^{8} passengers

Answer:

(C) 2.07 × 10^{7} passengers

Explanation:

20,700,000 Given

2.07 Move the decimal point 7 places to the left Remove the extra zeros

10,000,000 Divide the original number by the result from above.

10^{7} Write the answer as power of 10.

Write the product of the results from above

2.07 × 10^{7}

There are $2.07 \times 10^{7}$ passengers that travel by train

Question 11.

A quarter weighs about 0.025 pound. What is this weight written in scientific notation?

(A) 2.5 × 10^{-2} pound

(B) 2.5 × 10^{1} pound

(C) 2.5 × 10^{-1} pound

(D) 2.5 × 10^{2} pound

Answer:

(A) 2.5 × 10^{-2} pound

Explanation:

We need to express the number 0.025 in scientific notation (a × 10^{n}, where a is a number greater or equal to 1 but less than 10.

In that spirit we see we need to move the decimal point 2 places (to the right) so the power of 10 needs to be 2

(or -2). Since the given number is a number smaller than 1 the exponent in the power of 10 is going to be negative (-2).

0.025 = 2.5 × 10^{-2}

Question 12.

Which of the following is the number 3.0205 × 10^{-3} written in standard notation?

(A) 0.00030205

(B) 0.0030205

(C) 3.0205

(D) 3020.5

Answer:

The standard form of the number 3.0205 × 10 power of -3 = 0.00030205.

Option A is the correct answer.

Question 13.

A human fingernail has a thickness of about 4.2 × 10^{-4} meter. What is this width written in standard notation?

(A) 0.0000042 meter

(B) 0.000042 meter

(C) 0.00042 meter

(D) 0.0042 meter

Answer:

(C) 0.00042 meter

Explanation:

a We are given: 4.2 × 10^{-4} meter Since the power of 10 is a negative 4 we need to move the decimal place 4

places to the left to get a number written in scientific notation.

4.2 × 10^{-4} meter = 0.00042 meter Move decimal point and add placeholder zeros

b. Since 1 meter = 0.001 millimeter or 1 millimeter = 10^{3} meter we can calculate how many millimeters is 4.2 × 10^{-4} meter

4.2 × 10^{-4} meter = (4.2 × 10^{-4}) × 10^{3} millimeter = 0.42 millimeter

We can conclude that their measurements agree.

c. Since the thickness of a human fingernail is a very small number it is more appropriate to express it in milimeters than in meters (because a millimeter is 1000 times smaller than a meter).

**Gridded Response**

Question 14.

The square root of a number is -18. What is the other square root?

Answer:

A square root always has 2 numbers, one is positive and other is negative.

So, one number is -18 and the other number is 18

**Hot Tip!**

Underline key words given in the test question so you know for certain what the question is asking.

Question 15.

Jerome is writing a number in scientific notation. The number is greater than one million and less than ten million. What will be the exponent in the number Jerome writes?

Answer:

Given that Jerome is writing the number in the scientific notation.

The number is greater than one million and less than ten million.

The exponent in the number is

1 million = 10⁶

10 million = 10⁷

The number is > 10⁶ < 10⁷

The exponent number is 6.

Question 16.

Write the number 3.3855 × 10^{2} in standard notation.

Answer:

3.3855 × 10² in the standard form is 3.3855 × 100 = 338.55

**Texas Go Math Grade 8 Unit 1 Vocabulary Preview** **Answer Key**

**Use the puzzle to preview key vocabulary from this unit. Unscramble the circled letters to answer the riddle at the bottom of the page.**

Question 1.

TCREEFP

SEAQUR

Answer:

PERFECT

SQUARE

Question 2.

NOLRATAI

RUNMEB

Answer:

RATIONAL

NUMBER

Question 3.

PERTIANEG

MALCEDI

Answer:

PERTAINGE

DECIMAL

Question 4.

LAER

SEBMNUR

Answer:

REAL

NUMBERS

Question 5.

NIISICFTCE

OITANTON

Answer:

SCIENTIFIC

NOTATION

- Has integers as its square roots. (Lesson 1-1)
- Any number that can be written as a ratio of two integers. (Lesson 1-1)
- A decimal in which one or more digits repeat infinitely. (Lesson 1-1)
- The set of rational and irrational numbers. (Lesson 1-2)
- A method of writing very large or very small numbers by using powers of 10. (Lesson 2-1)

Q: What keeps a square from moving?

A: ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___!

Answer:

SQUARE ROOTS