## Envision Math Grade 8, Volume 1, Chapter 1: Real Number

**Page 56 Exercise 17 Answer**

We need to express 0.000000298 as a single digit times a power of ten rounded to the nearest ten millionth.

Rounded to the greatest place value, we get,

0.000000298=0.0000003

There are 7 zeros in the rounded number.

Thus, the value

3 x 10^{-7}

The value will be 3 x 10^{-7}

We need to explain how negative powers of 10 make small numbers easier to write and compare.

Negative powers of 10 are utilized in writing small quantities.

It will take up so much space to write very small quantities.

Here, the negative powers are used to move so many decimal spaces.

For example,

0.0000000000000004 can be written as 4 × 10^{-16}

This will save up so much space and enables easier calculation.

While writing small numbers, each negative power of 10 will be equal to one decimal place after the decimal point.

**Page 57 Exercise 1 Answer**

We need to explain what does the exponent in 10^{15} tell you about the value of the number.

The number given is 10^{15}

The exponent here is 15

This means that the number of zeroes present in the number is 15

Thus, the number will be more than 1,000,000,000,000,000

This denotes that the value 10^{15 }denotes quadrillions.

The exponent in 10^{15} tell us that the value of the number will be in quadrillions.

**Page 57 Exercise 1 Answer**

We need to explain how we can use the knowledge of powers of 10 to rewrite the number.

To rewrite any number into a power of 10, we need to use scientific notation.

In a scientific notation, it consists of two terms.

The first term will be of the number between one and ten. The second term will be the power of ten.

The exponent in the power of 10 denotes the number of zeroes present in the number.

If the exponent is negative, then it denotes the number of zeroes after the decimal point.

We can use the knowledge of powers of 10 to rewrite the number by placing the decimal after the first nonzero digit and by counting the number of digits after the decimal point.

**Page 58 Exercise 1 Answer**

Given that the height of Angel Falls, the tallest waterfall in the world, is 3,212 feet. We need to write this number in scientific notation.

The height of Angel falls is 3212ft

The number in scientific notation will have two factors.

The first factor will have values between one and ten.

The second factor will be the power of ten.

The number in scientific notation will be,

3212 = 3.212 x 10^{3}

The number in scientific notation will be 3.212 x 10^{3}

**Page 59 Exercise 3 Answer**

We need to write the numbers in standard form 9.225 × 10^{18}

The given number in scientific notation is 9.225 × 10^{18}

The number in scientific notation will have two factors.

The first factor will have values between one and ten.

The second factor will be the power of ten.

It can be written in standard form by counting the number of zeroes after the decimal point.

The number in standard form will be,

9.225 × 10^{18} = 9225000000000000000

The number in standard form will be, 9.225 × 10^{18} = 9225000000000000000

We need to write the numbers in standard form 6.3 × 10^{-8}

The given number in scientific notation is 6.3 × 10^{-8}

The number in scientific notation will have two factors.

The first factor will have values between one and ten.

The second factor will be the power of ten.

It can be written in standard form by counting the number of zeroes before the decimal point.

The number in standard form will be,

0.000000063

The number in standard form will be, 0.000000063

**Page 58 Exercise 1 Answer**

We need to explain why very large numbers have positive exponents when written in scientific notation.

We will use a power of10 to estimate a quantity that is either too big or too small to count.

Quantities that are neither too big nor too small can easily be represented.

We have to count the number of zeroes to write it as a power of 10

The big numbers will have a positive exponent.

The small numbers will have a negative exponent.

The large numbers have positive exponents when written in scientific notation because the number is large and the number of zeroes is more.

**Page 60 Exercise 1 Answer**

Scientific notations make calculations easier by writing very large or very small numbers into their compact form.

For example, 0.00000000005

This can be represented as 5 × 10^{-11}

In this way, we can rewrite large or small values into very compact forms using scientific notations.

Scientific notation is a convenient way to write very large numbers or very small numbers.

**Page 60 Exercise 2 Answer**

Given that, Taylor states that 2,800,000 in scientific notation is 2.8 × 10^{−6} because the number has six places to the right of the 2. We need to find whether the

Taylor’s reasoning is correct or not.

Given that, Taylor states that 2,800,000 in scientific notation is 2.8 × 10^{−6}

The number in scientific notation will have two factors.

The first factor will have values between one and ten.

The second factor will be the power of ten.

Here, we have six places to the right of the 2.

Thus, the value will be, 2.8 × 10^{6}

We will use negative exponent only when we have places to the left of the given number.

Taylor’s reasoning is wrong.

**Page 60 Exercise 4 Answer**

We need to write the numbers in scientific form 586,400,000

The given number in standard form is 586,400,000

The number in scientific notation will have two factors.

The first factor will have values between one and ten.

The second factor will be the power of ten.

The number in scientific form will be,

586,400,000 = 5.8 x 10^{8}

The number in the scientific form will be, 586,400,000=5.8 × 10^{8}

**Page 60 Exercise 7 Answer**

We need to write the number displayed on the calculator screen in standard form. The number displayed is 7.6E12

The given number in scientific notation is 7.6E12=7.6 × 10^{12}

The number in scientific notation will have two factors.

The first factor will have values between one and ten.

The second factor will be the power of ten.

It can be written in standard form by counting the number of zeroes after the decimal point.

The number in standard form will be,

7.6 x 10^{12} = 760000000000000

The number in standard form will be, 7.6 x 10^{12} = 7600000000000

**Page 61 Exercise 8 Answer**

Given that, the Sun is 1.5×108 kilometers from Earth. We need to write it as standard form.

The given number in scientific notation is 1.5 × 10^{8}

The number in scientific notation will have two factors.

The first factor will have values between one and ten.

The second factor will be the power of ten.

It can be written in standard form by counting the number of zeroes after the decimal point.

The number in standard form will be,

1.5 x 108 = 150000000

The number in standard form will be, 1.5 x 10^{8} = 150000000

**Page 61 Exercise 9 Answer**

Brenna wants an easier way to write 0.0000000000000000587. We need to write this in scientific notation.

The given number in standard form is 0.0000000000000000587

The number in scientific notation will have two factors.

The first factor will have values between one and ten.

The second factor will be the power of ten.

The number in the scientific form will be,

0.0000000000000000587=5.87 × 10^{−17}

The number in the scientific form will be, 0.0000000000000000587=5.87 × 10^{−17}

**Page 61 Exercise 10 Answer**

We need to check whether the number 23 × 10^{−8} is written in scientific notation or not.

The number in scientific notation will have two factors.

The first factor will have values between one and ten.

The second factor will be the power of ten.

Here, the given number is 23 × 10^{−8}

Thus, we are having two factors in this number.

Therefore, the given number is in scientific notation.

The given number 23 x 10^{−8 }is in scientific notation.

**Page 61 Exercise 12 Answer**

Given that, Simone evaluates an expression using her calculator. The calculator display is shown on the right. We need to express the number 5.2E−11 in standard form.

The number in scientific notation will have two factors.

The first factor will have values between one and ten.

The second factor will be the power of ten.

Here, the given number is 5.2E−11 that is 5.2E−11=5.2 × 10^{−11}

The given number in standard form will be,

5.2 × 10^{−11 }= 0.000000000052

The given number in standard form will be, 5.2 × 10^{−11 } = 0.000000000052

**Page 61 Exercise 13 Answer**

We need to write the given number 0.00001038 in scientific notation.

The number in scientific notation will have two factors.

The first factor will have values between one and ten.

The second factor will be the power of ten.

Here, the given number is 0.00001038

The given number in scientific notation will be,

0.00001038=1.038 × 10^{−5}

The given number in scientific notation will be,0.00001038=1.038 × 10^{−5}

**Page 61 Exercise 15 Answer**

Given that, Peter evaluates an expression using his calculator. The calculator display is shown at the right. We need to express the number 8.19E18 in standard form.

The number in scientific notation will have two factors.

The first factor will have values between one and ten.

The second factor will be the power of ten.

Here, the given number is 8.19E18 = 8.19 × 10^{18}

We need to count 18 decimal places to the right.

The given number in standard form will be,

8.19 x 10^{18} = 8190000000000000000

The given number in standard form will be, 8.19 × 10^{18} = 8190000000000000000

**Page 62 Exercise 16 Answer**

We need to describe what we should do first to write 5.871 × 10^{-7} in standard form.

The number in scientific notation will have two factors.

The first factor will have values between one and ten.

The second factor will be the power of ten.

Here, the given number is 5.871×10^{-7}

We need to count 7 decimal places to the left.

The given number in standard form will be,

5.871 × 10^{-7} = 0.0000005871

The given number in standard form will be, 5.871 × 10^{-7 }= 0.0000005871

The first step is to count 7 places to the left of the given decimal.

We need to express the number 5.871 × 10^{-7} in standard form.

The number in scientific notation will have two factors.

The first factor will have values between one and ten.

The second factor will be the power of ten.

Here, the given number is 5.871 × 10^{-7}

We need to count 7 decimal places to the left.

The given number in standard form will be,

5.817 x 10^{-7} = 0.0000005871

The given number in standard form will be, 5.871 x 10^{-7} = 0.0000005817

**Page 62 Exercise 17 Answer**

We need to express the given number 2.58×10^{-2} in standard form.

The number in scientific notation will have two factors.

The first factor will have values between one and ten.

The second factor will be the power of ten.

Here, the given number is 2.58 × 10^{-2}

We need to count 2 decimal places to the left.

The given number in standard form will be,

2.58 x 10^{-2} = 0.0258

The given number in standard form will be, 2.58 × 10^{-2} = 0.0258

**Page 62 Exercise 18 Answer**

Given that, At a certain point, the Grand Canyon is approximately 1,600,000 centimeters across. We need to express this number in scientific notation.

The number in scientific notation will have two factors.

The first factor will have values between one and ten.

The second factor will be the power of ten.

Here, the given number is 1,600,000

The given number in scientific notation will be,

1,600,000 = 1.6 x 10^{6}

The given number in scientific notation will be,1,600,000 = 1.6 × 10^{6}

**Page 62 Exercise 20 Answer**

We need to express the distance 4,300,000 meters using scientific notation in meters, and then in millimeters.

The number in scientific notation will have two factors.

The first factor will have values between one and ten.

The second factor will be the power of ten.

Here, the given number is 4,300,000

The given number in scientific notation will be,

4,300,000=4.3×10^{6} meters.

The given number in scientific notation in millimeters will be,

4,300,000 = 4.3 × 10^{6}

= 4.3 × 10^{6 }× 10^{3} × 10^{-3}

= 4.3 × 10^{9} × 10^{-3}

= 4.3 × 109 millimeters

The given number in scientific notation will be 4.3 × 10^{6 }meters or 4.3 × 10^{9} millimeters.

**Page 62 Exercise 21 Answer**

We need to find which of the given numbers is written in scientific notation.

The number in scientific notation will have two factors.

The first factor will have values between one and ten.

The second factor will be the power of ten.

The numbers in scientific notation are,

12 × 10^{6},6.89 × 10^{6}

Here, 12 and 6.89 doesn’t consider as in scientific notation form since they both are not too big or too small numbers.

Among the given, the following numbers are written in scientific notation.

(A) 12 × 10^{6}

(C) 6.89 × 10^{6}

**Page 62 Exercise 22 Answer**

Given that, Jeana’s calculator display shows the number to the right. We need to express this number in scientific notation.

The number shown in the calculator is 5.49E14

The number in scientific notation will have two factors.

The first factor will have values between one and ten.

The second factor will be the power of ten.

Here, the given number is 5.49E14

The given number in scientific notation will be,

5.49E14 = 5.49 × 10^{14}

The given number in scientific notation will be, 5.49E14 = 5.49 × 1014

Given that, Jeana’s calculator display shows the number to the right. We need to express this number in standard form.

The number shown in the calculator is 5.49E14

The number in scientific notation will have two factors.

The first factor will have values between one and ten.

The second factor will be the power of ten.

Here, the given number is 5.49E14 = 5.49 ×10^{14}

We need to count 14 places to the right of the decimal point.

The given number in standard form will be,

5.49 × 10^{14} = 549000000000000

The given number in standard form will be, 5.49 × 10^{14} = 549000000000000