Question

Help please on properties of exponents? will give a medal!

A rectangle has a length of the cube root of 81 inches and a width of 3 to the 2 over 3 power inches. Find the area of the rectangle.

a. 3 to the 2 over 3 power inches squared

b. 3 to the 8 over 3 power inches squared

c. 9 inches squared

d.9 to the 2 over 3 power inches squared

2.)Explain how the Quotient of Powers was used to simplify this expression.

2 to the fifth power, over 8 = 2 to the 2nd power

a.By finding the quotient of the bases to be one fourth and cancelling common factors

b. By finding the quotient of the bases to be one fourth and simplifying the expression

c. By simplifying 8 to 23 to make both powers base two and subtracting the exponents

d. By simplifying 8 to 23 to make both powers base two and adding the exponents

3.)the cube root of 2 to the seventh power

a. 2 to the 3 over 7 power

b. 2 to the 7 over 3 power

c. 2^21

d. 2^4,

Answer 1

Ques 1)

Option: C is the correct answer.

c. 9 inches square.

Ques 2)

Option: c

c. By simplifying 8 to 2^3 to make both powers base two and subtracting the exponents.

Ques 3)

Option: b

b. 2 to the 7 over 3 power

Explanation:

Ques 1)

A rectangle has a length of the cube root of 81 inches and a width of 3 to the 2 over 3 power inches.

i.e. let 'l' and 'b' denote the length and width of the rectangle.

i.e.
`l=\sqrt[3]{81}\\\\l=(3^4)^{(1)/(3)}\\\\l=3^{(4)/(3)`

since,

`(a^m)^n=a^(mn)`

and

`w=3^{(2)/(3)}`

Hence, the area of rectangle is given by:

`Area=l* w\\\\\\Area=3^{(1)/(3)}*3^{(4)/(3)}\\\\\\Area=3^({(1)/(3)+(4)/(3)})\\\\\\Area=3^{(6)/(3)}\\\\\\Area=3^2\\\\\\Area=9\ square\ inches`

As we know that:

`a^m* a^n=a^(m+n)`

Hence, Area=9 square inches.

Ques 2)

2 to the fifth power, over 8 = 2 to the 2nd power

i.e. we need to prove that:

`(2^5)/(8)=2^2`

As we know that:

`(2^5)/(8)=(2^5)/(2^3)=2^(5-3)=2^2`

( since

`(a^m)/(a^n)=a^(m-n)` )

Hence, the correct answer is: Option: c

Ques 3)

The cube root of 2 to the seventh power.

i.e.

`\sqrt[3]{2^7}\\\\=(2^7)^{(1)/(3)}\\\\=2^{(7)/(3)}`

Since,

`\sqrt[n]{x}=x^{(1)/(n)}`

and

`(a^m)^n=a^(mn)`

Hence, the correct answer is: option: b

Answer 2

Answer
1) c. 9 inches squared
2) c. By simplifying 8 to 23 to make both powers base two and subtracting the exponents
3) b. 2 to the 7 over 3 power

Explanation.
Question 1
The area of a rectangle is given by
area=l×w
=〖81〗^(1/3)×3^(2/3)
3^(4/3)×3^(2/3)=3^(4/3+2/3)
=3^(6/3)=3^2=9 inches squared

Question 2
Finding the 2^5/8=2^2
The steps for finding this is first to write 8 as 23. Then divide.
2^5/8=2^5/2^3
When dividing exponents with the same base you subtract the powers.
=2^(5-3)=2^2

Question 3
The question wants us to solve, ∛(3^7 ).the cube root of a number can be written as a power of a third.
So,
∛(3^7 )=3^(7×1/3 )
=3^(7/3)