Question

Question 1 (Multiple Choice)

(MC)

Rewrite the radical as a rational exponent.

the fourth root of 7 to the fifth power

7 to the 5 over 4 power

720

7

7 to the 4 over 5 power


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Question 2 (Multiple Choice)

(MC)

Rewrite the rational exponent as a radical by extending the properties of integer exponents.

2 to the 7 over 8 power, all over 2 to the 1 over 4 power

the eighth root of 2 to the fifth power

the fifth root of 2 to the eighth power

the square root of 2 to the 5 over 8 power

the fourth root of 2 to the sixth power


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Question 3 (Multiple Choice)

(MC)

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.

3 to the 2 over 3 power inches squared

3 to the 8 over 3 power inches squared

9 inches squared

9 to the 2 over 3 power inches squared


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Question 4 (Multiple Choice)

(MC)

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

5 to the fourth power, over 25 = 52

By simplifying 25 to 52 to make both powers base five and subtracting the exponents

By simplifying 25 to 52 to make both powers base five and adding the exponents

By finding the quotient of the bases to be one fifth and cancelling common factors

By finding the quotient of the bases to be one fifth and simplifying the expression


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Question 5 (Multiple Choice)

(MC)

Rewrite the rational exponent as a radical expression.

3 to the 2 over 3 power, to the 1 over 6 power

the sixth root of 3

the ninth root of 3

the eighteenth root of 3

the sixth root of 3 to the third power

Answer 1

Q1. The answer is 7 to the 5 over 4 power
Let's write the fourth root of 7 to the fifth power as a radical. 7 to the fifth power is 7⁵. The fourth root of 7 to the fifth power is
`\sqrt[4]{7^(5) }`.
Now, to rewrite it as a rational exponent, we will use the following:

`x^{ (m)/(n)}= \sqrt[n]{ x^(m) }`
Our radical is
`\sqrt[4]{7^(5) }` which means that n = 4, m = 5.
So, the rational exponent it will be:

`\sqrt[4]{ 7^(5)} =7^{ (5)/(4)}` which is the same as 7 to the 5 over 4 power


Q2. The answer is the eighth root of 2 to the fifth power.
Let's present 2 to the 7 over 8 power, all over 2 to the 1 over 4 power as a rational exponent.
2 to the 7 over 8 power is
`2^{ (7)/(8)}`
2 to the 1 over 4 power is
`2^{ (1)/(4) }`
2 to the 7 over 8 power, all over 2 to the 1 over 4 power is
`\frac{2^{ (7)/(8)}}{2^{ (1)/(4) }}`
Using the rule:
`(x^(a) )/( x^(b) )= x^(a-b)` we have:

`\frac{2^{ (7)/(8)}}{2^{ (1)/(4) }} =2^{ (7)/(8)- (1)/(4)} = 2^{ (7)/(8)- (2)/(8)}= 2^{ (7-2)/(8) } = 2^{ (5)/(8) }`
Since:
`x^{ (m)/(n)}= \sqrt[n]{ x^(m) }`, then n = 8, m = 5
Therefore

`2^{ (5)/(8) = \sqrt[8]{ 2^(5) }`


Q3. The answer is 9 inches squared.
The area of the rectangle (A) is
A = l · w (l - length, w - width).
It is given:
l = the cube root of 81 inches =
`\sqrt[3]{81}= \sqrt[3]{3^(4) } =3^{ (4)/(3) }`
w = 3 to the 2 over 3 =
`3^{ (2)/(3)}`

A =
`3^{ (4)/(3) } * 3^{ (2)/(3) }`
Since:
`x^(a) * x^(b)= x^(a+b)` then:
A =
`3^{ (4)/(3)+ (2)/(3)}= 3^{ (4+2)/(3) } = 3^{ (6)/(3) } = 3^(2)=9`


Q4. The answer is By simplifying 25 to 5² to make both powers base five and subtracting the exponents
5 to the fourth power, over 25 = 52 is
`( 5^(4))/(25)= 5^(2)`
Now, let's simplify 25 to 5²:

`( 5^(4))/( 5^(2) )=5^(2)`
Since
`(x^(a) )/( x^(b) )= x^(a-b)`, we will subtract the exponents:

`5^(4-2) = 5^(2)`
⇒
`5^(2) = 5^(2)`


Q5. The answer is the ninth root of 3
3 to the 2 over 3 power is
`3^{ (2)/(3) }`
3 to the 2 over 3 power, to the 1 over 6 power is
`(3^{ (2)/(3) } } )^{ (1)/(6) }`
Since
`(x^(a))^(b) =x ^(a*b)` then:

`(3^{ (2)/(3)}) ^{ (1)/(6) } = 3^{ (2)/(3) * (1)/(6) } = 3^{ (2)/(18) }= 3^{ (1)/(9) }`
Since:
`x^{ (m)/(n)}= \sqrt[n]{ x^(m) }`, then: n = 9, m = 1

`3^{ (1)/(9) }= \sqrt[9]{3^(1) } = \sqrt[9]{3}`