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

**Page 45 Exercise 1 Answer**

We need to determine whether the relationship shown for Set 1 is also true for Sets 2-5.

In the first set, the number of sit-ups they do is 64

In the next set, it will be 32 since it is half as many for each subsequent set.

For third set,16

The number of sit-ups is represented by the equivalent expression, 2^{7−n}

where n is the number of sets.

This expression is used to determine the number of sit-ups they do in each set.

Using this expression, finding the number of sit-ups in

Set 1 – 2^{7−n} = 2^{7−1} = 2^{6} = 64

Set 2 – 2^{7−n} = 2^{7−2} = 2^{5} = 32

Set 3 – 2^{7−n} = 2^{7−3} = 2^{4} = 16

Set 4 – 2^{7−n} = 2^{7−4} = 2^{3} = 8

Set 5 – 2^{7−n} = 2^{7−5} = 2^{2} = 4

The relationship shown for Set 1 is also true for Sets 2-5.

**Page 45 Exercise 1 Answer**

Given that, Calvin and Mike do sit-ups when they work out. They start with 64 sit-ups for the first set and do half as many for each subsequent set.

We need to explain how could we determine the number of sit-up sets Calvin and Mike do.

In the first set, the number of sit-ups they do is 64

In the next set, it will be 32 since it is half as many for each subsequent set.

For third set,16

We know that, 64=2^{7}

For each set, it is reduced as half.

The number of sit-ups is represented by the equivalent expression, 2^{7−n}

where n is the number of sets.

This expression is used to determine the number of sit-ups they do in each set.

The expression is used to determine the number of sit-ups they do in each set is 2^{7−n}

**Page 46 Question 1 Answer**

The zero exponent property states that when the exponent is zero while the base is non-zero, then the result will be one.

This means that, x^{0} = 1 where x ≠ 0

Example, 15^{0} = 1

The negative exponent property states that when the exponent is negative, we need to put the number as a denominator in a fraction by changing the exponent into positive.

This means that, \(x^{-n}=\frac{1}{x^n}\) where x ≠ 0

Example, \(15^{-2}=\frac{1}{15^2}\)

The zero exponent property states that when the exponent is zero while the base is non-zero, then the result will be one.

The negative exponent property states that when the exponent is negative, we need to put the number as a denominator in a fraction by changing the exponent into positive.

**Page 46 Exercise 1 Answer**

We need to evaluate (−7)^{0}

The zero exponent property states that when the exponent is zero while the base is non-zero, then the result will be one.

The value of (−7)^{0 }= 1

We need to evaluate (43)^{0}

We need to evaluate 1^{0
}

The value of 1^{0} = 1

We need to evaluate (0.5)^{0}

The value of (0.5)^{0} = 1

**Page 47 Exercise 3 Answer**

We need to evaluate \(\frac{1}{5^{-3}}\)

The negative exponent property states that when the exponent is negative, we need to put the number as a denominator in a fraction by changing the exponent into positive.

This means that, \(x^{-n}=\frac{1}{x^n} \text { where } x \neq 0\)

**Page 46 Exercise 1 Answer**

We need to evaluate why 2(7^{0}) = 2

The value 7^{0} = 1. This makes 2(7^{0}) = 2

**Page 48 Exercise 1 Answer**

This means that, x^{0} = 1 where x ≠ 0

Example, 17^{0} = 1

This means that, \(x^{-n}=\frac{1}{x^n} \text { where } x \neq 0\)

Example, \(5^{-2}=\frac{1}{5^2}\)

**Page 48 Exercise 2 Answer**

We need to describe what does the negative exponent mean in the expression 9^{−12}

The negative exponent means that it can be written as a fraction with a positive exponent using the negative exponent property.

**Page 48 Exercise 4 Answer**

We need to simplify the expression 1999999^{0}

**Page 48 Exercise 6 Answer**

We need to find the value of 27x^{0}y^{−2} when the value of x=4 and y=3

The given expression is 27x^{0}y^{−2}

Using the zero and negative exponents property, the expression will be,

The value of 27x^{0}y^{−2} = 3

**Page 49 Exercise 7 Answer**

We need to complete the given table to find the value of a nonzero number raised to the power of 0.

The given table is,

By evaluating and completing the given table, we get,

4^{4} = 256

The completed table is,

**Page 49 Exercise 8 Answer**

We need to complete the given table to find the value of a nonzero number raised to the power of 0.

The given table is,

By evaluating and completing the given table, we get,

(−2)^{4} = −2 × −2 × −2 × −2 = 16

(−2)^{3} = −2 × −2 × −2 = −8

(−2)^{2} = −2 × −2 = 4

(−2)^{1} = −2

(−2)^{0} = 1

The completed table is,

**Page 49 Exercise 9 Answer**

We need to simplify the given expression (−3.2)^{0}

This means that, x^{0} = 1 where x ≠ 0

Using this property, we get,

(−3.2)^{0} = 1

The value of (−3.2)^{0 }= 1

We need to write two expressions equivalent to the given expression (−3.2)^{0}. Also, explain why the three expressions are equivalent.

Thus the value of the given expression is (−3.2)^{0} = 1

Therefore, the two expressions equivalent to the given expression is, 5^{0 }and (−15)^{0}

This is because according to zero exponent property, anything to the power of 0 will result in the number one.

The two expressions equivalent to the given expression is 5^{0} and (−15)^{0}.

The three expressions are equivalent since according to zero exponent property, any number to the power of 0 will result in 1.

**Part 49 Exercise 10 Answer**

We need to simplify the expression 12x^{0}(x^{−4}) when the value of x = 6

Thus, using these properties, we get,

The value of \(12 x^0\left(x^{-4}\right)=\frac{1}{108}\)

We need to simplify the expression 14(x^{−2}) when the value of x = 6

This means that, \(x^{-n}=\frac{1}{x^n} \text { where } x \neq 0\)

Using this property, we get,

This value of \(14\left(x^{-2}\right)=\frac{7}{18}\)

**Page 49 Exercise 12 Answer**

We need to compare the values using <, > or =

The gives value is \(\left(\frac{1}{4}\right)^0 ? 1\)

Using this property, we get,

\(\left(\frac{1}{4}\right)^0=1\)Thus, the given value becomes, \(\left(\frac{1}{4}\right)^0=1\)

The given value becomes,\(\left(\frac{1}{4}\right)^0=1\)

**Page 49 Exercise 13 Answer**

We need to rewrite each expression using a positive exponent. The given expression is 9^{−4}

Rewriting the expression using a positive exponent, we get,\(9^{-4}=\frac{1}{9^4}\)

**Page 49 Exercise 14 Answer**

We need to rewrite each expression using a positive exponent.

The given expression is \(\frac{1}{2^{-6}}\)

This means that, \(x^{-n}=\frac{1}{x^n} \text { where } x \neq 0\)

Using this property, we get,

\(\frac{1}{2^{-6}}=2^6\)Rewriting the expression using a positive exponent, we get, \(\frac{1}{2^{-6}}=2^6\)

**Page 49 Exercise 15 Answer**

We need to evaluate 9y^{0} when y=3

This means that, x^{0} = 1 where x ≠ 0

Using this property, we get,

9y^{0} = 9(1)

= 9

The value of 9y^{0} = 9

Given: 9y^{0}

We need to find whether the value of the given expression will vary depending on y

This means that, x^{0} = 1 where x ≠ 0

Using this property, we get,

9y^{0 }= 9(1) = 9

The result obtained will not have any relation with the value of y

The value of the given expression does not vary depending on the value of y

**Page 50 Exercise 16 Answer**

We need to simplify the expression −5x^{−4} when x = 4

This means that, \(x^{-n}=\frac{1}{x^n} \text { where } x \neq 0\)

Using this property, we get,

We need to simplify the expression 7x^{−3} when x=4

This means that, \(x^{-n}=\frac{1}{x^n} \text { where } x \neq 0\)

Using this property, we get,

The value of \(7 x^{-3}=\frac{7}{64}\)

**Page 50 Exercise 17 Answer**

We need to evaluate the expression (−3)^{−8} and −3^{−8}

The given expressions are (−3)^{−8} and −3^{−8}

Evaluating using negative exponents property, we get,

The value of \((-3)^{-8}=\frac{1}{6561} \text { and }-3^{-8}=\frac{-1}{6561}\)

We need to evaluate are (−3)^{−9} and −3^{−9}

The given expressions are (−3)^{−9} and −3^{−9}

Evaluating using negative exponents property, we get,

The values of \((-3)^{-9}=\frac{-1}{19683} \text { and }-3^{-9}=\frac{-1}{19683}\)

**Page 50 Exercise 19 Answer**

We have to simplify the expression. We need to assume that x is nonzero. Our answer should have only positive exponents.

The expression is x^{−10}x^{6}

Using this property, we get,

The value of \(x^{-10} x^6=\frac{1}{x^4}\)

**Page 50 Exercise 20 Answer**

We need to find whether the value of the expression \(\left(\frac{1}{4^{-3}}\right)^{-2}\) is greater than 1, equal to 1, or less than 1.

Using this property, we get,

The value is less than 1.

The value of \(\left(\frac{1}{4^{-3}}\right)^{-2}=\frac{1}{4096}\) which is less than the number 1.

Given that, If the value of the expression is greater than 1, we need to show how you can change one sign to make the value less than 1. If the value is less than 1, we need to show how you can change one sign to make the value greater than 1 . If the value is equal to 1, we need to show how you can make one change to make the value not equal to 1.

The given expression is,\(\left(\frac{1}{4^{-3}}\right)^{-2}\)

It will be equal to the value \(\left(\frac{1}{4^{-3}}\right)^{-2}=\frac{1}{4096}\)

The value is less than the number 1.

In order to make the value greater than 1, change one of the signs of the expression as follows,

\(\left(\frac{1}{4^{-3}}\right)^2=\left(4^3\right)^2=64^2=4096\)In this way, we can make it greater than one.

The sign was changed from \(\left(\frac{1}{4^{-3}}\right)^{-2} \text { to }\left(\frac{1}{4^{-3}}\right)^2\) to make it greater than one.

**Page 50 Exercise 22 Answer**

We need to find which expressions have values less than 1 when x = 4 from the given expressions.

Using these properties, solving the given, we get,

The values which are less than 1 are,

\(\frac{x^0}{3^2}\) \(3 x^{-4}\)