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

**Page 39 Exercise 1 Answer**

Given that, One band’s streaming video concert to benefit a global charity costs $1.00 to view.The first day, the concert got 2,187 views. The second day, it got about three times as many views. On the third day, it got 3 times as many views as on the second day. If the trend continues, we need to determine how much money will the band raise on the day 7.

Using the prime factorization method, the money raised on the last day will be, \(2187 \times 3^7\)

Also, write the number 2187 in terms of prime factorization, we get,

The money will the band raise on the day 7 will be 3^{14
}

**Page 39 Exercise 1 Answer**

We need to use prime factorization to write an expression equivalent to the amount of money raised by the band on the last day of the week.

As we know that the money raised on day one is 2187, for each consecutive days the money is getting tripled.

Thus calculating the money raised for each day, we get,

2187 × 3 = 6561

6561 × 3 = 19683

19683 × 3 = 59049

59049 × 3 = 177147

177147 × 3 = 531441

531441 × 3 = 1594323

Thus, on the day 7,the money raised will be $1,594,323

The amount of money raised by the band on the last day of the week will be $1,594,323

**Page 40 Question 1 Answer**

We can write equivalent expressions using the properties of integer exponents to simplify the expressions.

Also, we can combine the expressions using these properties.

We can distribute the exponents using these properties.

We can combine the exponents together if the bases are the same.

We can deduce the exponents if the bases are the same and if they are in the division.

In this way, the properties are helpful in simplifying the expressions.

The properties of integer exponents help you write equivalent expressions by combining, distributing and simplifying the powers and bases.

**Page 40 Exercise 1 Answer**

Given that, the local zoo welcomed a newborn African elephant that weighed 3^{4} It is expected that at adulthood, the newborn elephant will weigh approximately 3^{4} times as much as its birth weight. We need to find the expression that represents the expected adult weight of the newborn elephant.

The weight of the newborn will be 3^{4}

The weight of the adult African elephant will be 3^{4} × 3^{4}

We can combine the exponents together if the bases are the same using the properties of integer exponents.

Thus, the weight of the adult will be,

3^{4} x 3^{4} = 3^{4+4}

= 3^{8}

The weight of the adult African elephant will be 3^{8}

**Page 41 Exercise 2 Answer**

Write equivalent expressions of (7^{3})^{2} using the properties of exponents.

The power of powers property implies to multiply the powers to find the power of powers.

Therefore, using the power of powers property, we get,

Write equivalent expressions of (4^{5})^{3} using the properties of exponents.

The power of powers property implies multiplying the powers to find the power of powers.

Therefore, using the power of powers property, we get,

Write equivalent expressions of 9^{4}×8^{4} using the properties of exponents.

The power of products property implies multiplying the bases when the exponents are same.

Therefore, using the power of products property, we get,

Write equivalent expressions of \(\frac{8^9}{8^3}\) using the properties of exponents.

The Quotient of powers property implies subtracting the powers when the bases are the same in a division.

**Page 40 Exercise 1 Answer**

We need to explain why the Product of Powers Property makes mathematical sense.

The properties of integer exponents help you write equivalent expressions by combining, distributing and simplifying the powers and bases.

**Page 42 Exercise 2 Answer**

When we are writing an equivalent expression for 2^{3}⋅2^{4}, we need to find how many times would you write 2 as a factor.

The bases are the same, thus adding the powers together we get,

We have to write 2 as a factor 7 times.

**Page 42 Exercise 5 Answer**

We need to write an equivalent expression for 7^{12}⋅7^{4}

The bases are the same, thus adding the powers together we get,

The equivalent expression is 7^{16}

**Page 42 Exercise 6 Answer**

Write equivalent expressions of (8^{2})^{4} using the properties of exponents.

The power of powers property implies multiplying the powers to find the power of powers.

Therefore, using the power of powers property, we get,

The equivalent expression is 8^{8}

**Page 42 Exercise 8 Answer**

We need to write an equivalent expression for \(\frac{18^9}{18^4}\)

The bases are the same in the given fraction, thus subtracting the exponents we get,

The equivalent expression is 18^{5}

**Page 43 Exercise 16 Answer**

We need to write an equivalent for the given expression \(\frac{3^{12}}{3^3}\)

If the bases are the same in a fraction, then we need to subtract the exponents together to get the result.

The equivalent expression is 3^{9}

**Page 43 Exercise 17 Answer**

We need to write an equivalent expression for the given expression 4^{5} . 4^{2}

If the bases are the same, then we need to add the exponents together to get the result.

The equivalent expression is 4^{7}

**Page 43 Exercise 18 Answer**

We need to write an equivalent expression for the given expression 6^{4} . 2^{4}

The power of products property implies multiplying the bases when the exponents are the same.

Therefore, using the power of products property, we get,

The equivalent expression is 12^{4}

**Page 44 Exercise 20 Answer**

We need to find whether the expression 8 x 8^{5} is equivalent to (8 x 8)^{5} or not.

The expression 8 x 8^{5} is not equivalent to (8 x 8)^{5}

**Page 44 Execise 21 Answer**

We need to find whether the expression (3^{2})^{3} is equivalent to (3^{3})^{2}

The power of powers property implies multiplying the powers to find the power of powers.

The expression (3^{2})^{3} is equivalent to (3^{3})^{2}

**Page 44 Exercise 22 Answer**

We need to find whether the expression \(\frac{3^2}{3^3}\) is equivalnet to \(\frac{3^3}{3^2}\) or not.

If the bases are the same in a fraction, then we need to subtract the exponents together to get the result.

Thus, we get,

The expression \(\frac{3^2}{3^3}\) is not equivalent to \(\frac{3^3}{3^2}\)

**Page 44 Exercise 23 Answer**

We need to find the width of the rectangle written as an exponential expression.

The width of the rectangle written as an exponential expression is w = 10^{1}m

**Page 44 Exercise 24 Answer**

We need to simplify the given expression \(\left(\left(\frac{1}{2}\right)^3\right)^3\)

The power of powers property implies multiplying the powers to find the power of powers.

The simplified expression is \(\left(\frac{1}{2}\right)^9\)

**Page 44 Exercise 25 Answer**

we need to use a property of exponents to write (3b)^{5} as a product of powers.

This property states that, if the bases are the same, then we need to add the exponents together to get the result.

The given expression is (3b)^{5}

Using the property we write it as,

(3b)^{5} = (3b)^{2} x (3b)^{3}

Writing the given expression as a product of powers, we get,

(3b)^{5} = (3b)^{2} x (3b)^{3}

**Page 44 Exercise 26 Answer**

We need to simplify the given expression 4^{5} . 4^{10}

If the bases are the same, then we need to add the exponents together to get the result.

The simplified expression is 4^{15}

**Page 44 Exercise 27 Answer**

Given that, your teacher asks the class to evaluate the expression (2^{3})^{1}. Your classmate gives an incorrect answer of 16.

We need to evaluate the expression.

The power of powers property implies multiplying the powers to find the power of powers.

The correct answer is (2^{3})^{1} = 8

Your teacher asks the class to evaluate the expression (2^{3})^{1}. Your classmate gives an incorrect answer of 16. We need to find which of the following error they made.

(A) Your classmate divided the exponents.

(B) Your classmate multiplied the exponents.

(c) Your classmate added the exponents.

(D) Your classmate subtracted the exponents.

The power of powers property implies multiplying the powers to find the power of powers.

Thus,

(2^{3})^{1} = 2^{3×1}

= 2^{4} = 16

They liked added the exponents.

If they added the exponents, the result will be,

(2^{3})^{1} = 2^{3+1} = 2^{4} = 16

This will lead to the incorrect answer.

The likely error is that (c) Your classmate added the exponents.