Question

1. Which of the following is equivalent to x5y2/xy2 when x ≠ 0 and y ≠ 0?

Answer 1

Q1. The answer is D. x4

Let's first rewrite the expression:
x⁵y²/xy² = x⁵/x * y²/y²

Using the rule xᵃ/xᵇ = x(ᵃ⁻ᵇ), we can write the expression as following:
x⁵y²/xy² = x⁵/x * y²/y² = x⁵⁻¹ * y²⁻² = x⁴ * y⁰ = x⁴ * 1 = x⁴

Thus, the correct answer is D.


Q2. The answer is A. 5(5^3/2/5)^2

125 in the form of exponent is 5³.
125 = 5³
Now, let's calculate all choices. 
 
The rules we will use are:
xᵃ * xᵇ = x(ᵃ⁺ᵇ)
xᵃ/xᵇ = x(ᵃ⁻ᵇ)
(xᵃ)ᵇ = xᵃ*ᵇ

A. 5(5³/2/5)² = 5 * (5³ * 5/2)²
                      = 5 * (5³⁺¹/2)²
                      = 5 * (5⁴/2)²
                      = 5 * (5⁴)²/(2)²
                      = 5 * 5⁴*²/4
                      = 5 * 5⁸ / 4
                      = 5¹⁺⁸ / 4
                      = 5⁹/4
                      ≠ 5³ ≠ 125

B. (5³/5⁴)⁻³ = (5³⁻⁴)⁻³
                   = (5⁻¹)⁻³
                   = 5⁽⁻¹⁾*⁽⁻³⁾
                   = 5³
                   = 125

C. 5⁻²/5⁻⁵  = 5⁽⁻²⁾⁻⁽⁻⁵) 
                  = 5⁽⁻²⁾⁺⁵
                  = 5³
                  = 125

D. 5(5⁵/5³) = 5 * 5⁵⁻³
                  = 5 * 5²
                  = 5¹⁺²
                  = 5³
                  = 125

Therefore, the only expression that is not equal to 125 is A.


Q3. The answer is 63x5

Let's check all choices
The rules we will use are:
xᵃ * xᵇ = x(ᵃ⁺ᵇ)
xᵃ/xᵇ = x(ᵃ⁻ᵇ)
(xᵃ)ᵇ = xᵃ*ᵇ

A. 6³x 
6³x/6x⁵ = 6³/6 * x/x⁵
             = 6³⁻¹ * x¹⁻⁵
             = 6²x⁻⁴
             = 36x⁻⁴
             ≠ 36

B. 6³x⁵ 
6³x⁵/6x⁵ = 6³/6 * x⁵/x⁵
              = 6³⁻¹ * x⁵⁻⁵
              = 6² * x⁰
              = 36 * 1
              = 36

C. 6x⁵ 
6x⁵/6x⁵ = 1
             ≠ 36

D. 6⁷x⁵
6⁷x⁵/6x⁵ = 6⁷/6 * x⁵/x⁵
              = 6⁷⁻¹ * x⁵⁻⁵
              = 6⁶ * x⁰
              = 46656 * 1
              ≠ 36

Therefore, the correct choice is B.


Q4. The answer is

We will use the rule: xᵃ/xᵇ = x(ᵃ⁻ᵇ)

5.4 x 10¹²/1.2 x 10³ = 5.4 / 1.2 x 10¹²/10³
                               = 4.5 x 10¹²⁻³
                               = 4.5 x 10⁹


Q5. The answer is B. To subtract powers with the same base, divide the exponents

Some of the rules regarding operations with exponents are:
xᵃ/xᵇ = x(ᵃ⁻ᵇ) - choice A
xᵃ * xᵇ = x(ᵃ⁺ᵇ) - choice C
(xᵃ)ᵇ = xᵃ*ᵇ - choice D

Through the process of elimination, choice B is not true.