Pre-AP Algebra 1 Practice Test & PDF Worksheets - Free Printable

Let's solve each problem step by step using the properties of integer exponents. The key rules we'll use are:

- $\frac{a^m}{a^n} = a^{m-n}$ (when dividing like bases)
- $\frac{-a}{-b} = \frac{a}{b}$ (negative divided by negative is positive)
- $\frac{a}{-b} = -\frac{a}{b}$ (positive divided by negative is negative)
- Coefficients are simplified as regular fractions.
- Variables with exponents are simplified using exponent rules.

---

1. Simplify:

$$
\frac{12x^7y^4}{-6x^3y^5}
$$

Step 1: Simplify coefficients:
$$
\frac{12}{-6} = -2
$$

Step 2: Simplify variables:
- $x^7 / x^3 = x^{7-3} = x^4$
- $y^4 / y^5 = y^{4-5} = y^{-1}$

So:
$$
-2x^4y^{-1}
$$

But since we want positive exponents, write $y^{-1} = \frac{1}{y}$, so:
$$
\boxed{-\frac{2x^4}{y}}
$$

---

2. Simplify:

$$
\frac{-6x^6y^6}{-2x^4y^7}
$$

Step 1: Coefficients:
$$
\frac{-6}{-2} = 3
$$

Step 2: Variables:
- $x^6 / x^4 = x^{6-4} = x^2$
- $y^6 / y^7 = y^{6-7} = y^{-1} = \frac{1}{y}$

So:
$$
\boxed{\frac{3x^2}{y}}
$$

---

3. Simplify:

$$
\frac{-27x^7y^2}{-9x^6y^5}
$$

Step 1: Coefficients:
$$
\frac{-27}{-9} = 3
$$

Step 2: Variables:
- $x^7 / x^6 = x^{1} = x$
- $y^2 / y^5 = y^{2-5} = y^{-3} = \frac{1}{y^3}$

So:
$$
\boxed{\frac{3x}{y^3}}
$$

---

4. Simplify:

$$
\frac{-32x^6y^5}{-8x^7y}
$$

Step 1: Coefficients:
$$
\frac{-32}{-8} = 4
$$

Step 2: Variables:
- $x^6 / x^7 = x^{6-7} = x^{-1} = \frac{1}{x}$
- $y^5 / y^1 = y^{5-1} = y^4$

So:
$$
\boxed{\frac{4y^4}{x}}
$$

---

5. Simplify:

$$
\frac{42x^4y}{-7xy^3}
$$

Step 1: Coefficients:
$$
\frac{42}{-7} = -6
$$

Step 2: Variables:
- $x^4 / x = x^{4-1} = x^3$
- $y / y^3 = y^{1-3} = y^{-2} = \frac{1}{y^2}$

So:
$$
\boxed{-\frac{6x^3}{y^2}}
$$

---

6. Simplify:

$$
\frac{-25x^4y^3}{-5x^3y^5}
$$

Step 1: Coefficients:
$$
\frac{-25}{-5} = 5
$$

Step 2: Variables:
- $x^4 / x^3 = x^{1} = x$
- $y^3 / y^5 = y^{-2} = \frac{1}{y^2}$

So:
$$
\boxed{\frac{5x}{y^2}}
$$

---

7. Show two ways to use a calculator to find the value of $\frac{3^6}{3^2}$

We know from exponent rules:
$$
\frac{3^6}{3^2} = 3^{6-2} = 3^4 = 81
$$

#### Method 1: Direct calculation
Enter into calculator:
```
3 ^ 6 ÷ 3 ^ 2
```
Which computes $729 ÷ 9 = 81$

#### Method 2: Use exponent subtraction rule
First compute $3^4$ directly:
```
3 ^ 4
```
This gives $81$

Alternatively, you can also do:
- Compute numerator: $3^6 = 729$
- Compute denominator: $3^2 = 9$
- Then divide: $729 ÷ 9 = 81$

So two valid calculator methods:
1. Enter `3^6 / 3^2`
2. Enter `(3^6) ÷ (3^2)` or compute separately and divide.

Answer: $\boxed{81}$

---

8. Write $b^{12}$ as a quotient form $\frac{b^m}{b^n}$ in four different ways. Use only positive exponents.

We need to express $b^{12}$ as $\frac{b^m}{b^n}$ such that:
$$
\frac{b^m}{b^n} = b^{m-n} = b^{12}
\Rightarrow m - n = 12
$$

So pick values of $m$ and $n$ such that $m > n$, and $m - n = 12$. Here are four examples:

1. $m = 14, n = 2$:
$\frac{b^{14}}{b^2} = b^{12}$

2. $m = 15, n = 3$:
$\frac{b^{15}}{b^3} = b^{12}$

3. $m = 20, n = 8$:
$\frac{b^{20}}{b^8} = b^{12}$

4. $m = 12, n = 0$:
$\frac{b^{12}}{b^0} = b^{12}$ (since $b^0 = 1$)

All are valid because they use positive exponents and result in $b^{12}$.

✔ So four possible answers:
$$
\boxed{
\frac{b^{14}}{b^2},\quad
\frac{b^{15}}{b^3},\quad
\frac{b^{20}}{b^8},\quad
\frac{b^{12}}{b^0}
}
$$

(Note: $b^0 = 1$, so this is acceptable.)

---

✔ Final Answers Summary:

1. $-\dfrac{2x^4}{y}$
2. $\dfrac{3x^2}{y}$
3. $\dfrac{3x}{y^3}$
4. $\dfrac{4y^4}{x}$
5. $-\dfrac{6x^3}{y^2}$
6. $\dfrac{5x}{y^2}$
7. Two calculator methods:
- Type: `3^6 / 3^2`
- Or: Compute $3^6 = 729$, $3^2 = 9$, then $729 ÷ 9 = 81$
Answer: $\boxed{81}$
8. Four expressions:
$\boxed{\dfrac{b^{14}}{b^2},\ \dfrac{b^{15}}{b^3},\ \dfrac{b^{20}}{b^8},\ \dfrac{b^{12}}{b^0}}$

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