1. Which of the following is equivalent to [tex]$x^5y^2/xy^2$[/tex] when [tex]$x \neq 0$[/tex] and [tex]$y \neq 0$[/tex]?

A. [tex]$x^6y^5$[/tex]
B. [tex]$x^5y$[/tex]
C. [tex]$x^4y$[/tex]
D. [tex]$x^4$[/tex]

2. Which expression is NOT equal to 125?

A. [tex]$5\left(\frac{5^3}{2/5}\right)^2$[/tex]
B. [tex]$\left(\frac{5^3}{5^4}\right)^{-3}$[/tex]
C. [tex]$\frac{5^{-2}}{5^{-5}}$[/tex]
D. [tex]$5\left(\frac{5^5}{5^3}\right)$[/tex]

3. A fraction reduces to 36. If its denominator is [tex]$6x^5$[/tex], what is its numerator?

A. [tex]$63x$[/tex]
B. [tex]$63x^5$[/tex]
C. [tex]$6x^5$[/tex]
D. [tex]$67x^5$[/tex]

4. Which is the correct simplification of [tex]$5.4 \times 10^{12}/1.2 \times 10^3$[/tex] written in scientific notation?

A. [tex]$4.5 \times 10^7$[/tex]
B. [tex]$4.5 \times 10^9$[/tex]
C. [tex]$45 \times 10^6$[/tex]
D. [tex]$6.48 \times 10^7$[/tex]

5. Which of the following statements is not true regarding operations with exponents?

A. To divide powers with the same base, subtract the exponents.
B. To subtract powers with the same base, divide the exponents.
C. To multiply powers with the same base, add the exponents.
D. To raise a power to a power, multiply the exponents.

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.