Simplifying Exponent Expressions worksheet with algebra problems for students to solve.

Problem: Simplifying Exponent Expressions

The task involves simplifying a series of expressions involving exponents. We will use the following exponent rules:
1. Power of a Power Rule: \((x^a)^b = x^{a \cdot b}\)
2. Product of Powers Rule: \(x^a \cdot x^b = x^{a + b}\)
3. Quotient of Powers Rule: \(\frac{x^a}{x^b} = x^{a - b}\)
4. Negative Exponent Rule: \(x^{-a} = \frac{1}{x^a}\)

Let's solve each expression step by step.

---

#### Expression 1:
\[
\frac{3x^8 (x^2)^6}{3x^5 (x^2)^2}
\]

1. Simplify the powers using the Power of a Power Rule:
\[
(x^2)^6 = x^{2 \cdot 6} = x^{12}, \quad (x^2)^2 = x^{2 \cdot 2} = x^4
\]
So the expression becomes:
\[
\frac{3x^8 \cdot x^{12}}{3x^5 \cdot x^4}
\]

2. Use the Product of Powers Rule in the numerator and denominator:
\[
3x^8 \cdot x^{12} = 3x^{8+12} = 3x^{20}, \quad 3x^5 \cdot x^4 = 3x^{5+4} = 3x^9
\]
So the expression is now:
\[
\frac{3x^{20}}{3x^9}
\]

3. Simplify the coefficients and use the Quotient of Powers Rule:
\[
\frac{3}{3} = 1, \quad \frac{x^{20}}{x^9} = x^{20-9} = x^{11}
\]
Therefore:
\[
\frac{3x^8 (x^2)^6}{3x^5 (x^2)^2} = x^{11}
\]

---

#### Expression 2:
\[
4x^{-4} (x^4)^4 x^{-1}
\]

1. Simplify the power using the Power of a Power Rule:
\[
(x^4)^4 = x^{4 \cdot 4} = x^{16}
\]
So the expression becomes:
\[
4x^{-4} \cdot x^{16} \cdot x^{-1}
\]

2. Use the Product of Powers Rule to combine the exponents:
\[
x^{-4} \cdot x^{16} \cdot x^{-1} = x^{-4 + 16 - 1} = x^{11}
\]
Therefore:
\[
4x^{-4} (x^4)^4 x^{-1} = 4x^{11}
\]

---

#### Expression 3:
\[
\frac{2x^3 (x^2)^4}{7x^{-3} (x^{-3})^{-3}}
\]

1. Simplify the powers using the Power of a Power Rule:
\[
(x^2)^4 = x^{2 \cdot 4} = x^8, \quad (x^{-3})^{-3} = x^{-3 \cdot -3} = x^9
\]
So the expression becomes:
\[
\frac{2x^3 \cdot x^8}{7x^{-3} \cdot x^9}
\]

2. Use the Product of Powers Rule in the numerator and denominator:
\[
2x^3 \cdot x^8 = 2x^{3+8} = 2x^{11}, \quad 7x^{-3} \cdot x^9 = 7x^{-3+9} = 7x^6
\]
So the expression is now:
\[
\frac{2x^{11}}{7x^6}
\]

3. Simplify the coefficients and use the Quotient of Powers Rule:
\[
\frac{2}{7}, \quad \frac{x^{11}}{x^6} = x^{11-6} = x^5
\]
Therefore:
\[
\frac{2x^3 (x^2)^4}{7x^{-3} (x^{-3})^{-3}} = \frac{2x^5}{7}
\]

---

#### Expression 4:
\[
7x^{-2} (x^5)^5
\]

1. Simplify the power using the Power of a Power Rule:
\[
(x^5)^5 = x^{5 \cdot 5} = x^{25}
\]
So the expression becomes:
\[
7x^{-2} \cdot x^{25}
\]

2. Use the Product of Powers Rule to combine the exponents:
\[
x^{-2} \cdot x^{25} = x^{-2 + 25} = x^{23}
\]
Therefore:
\[
7x^{-2} (x^5)^5 = 7x^{23}
\]

---

#### Expression 5:
\[
\frac{x^4 (x^6)^{-2}}{x^3 (x^4)^2}
\]

1. Simplify the powers using the Power of a Power Rule:
\[
(x^6)^{-2} = x^{6 \cdot -2} = x^{-12}, \quad (x^4)^2 = x^{4 \cdot 2} = x^8
\]
So the expression becomes:
\[
\frac{x^4 \cdot x^{-12}}{x^3 \cdot x^8}
\]

2. Use the Product of Powers Rule in the numerator and denominator:
\[
x^4 \cdot x^{-12} = x^{4-12} = x^{-8}, \quad x^3 \cdot x^8 = x^{3+8} = x^{11}
\]
So the expression is now:
\[
\frac{x^{-8}}{x^{11}}
\]

3. Use the Quotient of Powers Rule:
\[
\frac{x^{-8}}{x^{11}} = x^{-8-11} = x^{-19}
\]
Using the Negative Exponent Rule:
\[
x^{-19} = \frac{1}{x^{19}}
\]
Therefore:
\[
\frac{x^4 (x^6)^{-2}}{x^3 (x^4)^2} = \frac{1}{x^{19}}
\]

---

#### Expression 6:
\[
6x^6 (x^5)^4 x^{-2}
\]

1. Simplify the power using the Power of a Power Rule:
\[
(x^5)^4 = x^{5 \cdot 4} = x^{20}
\]
So the expression becomes:
\[
6x^6 \cdot x^{20} \cdot x^{-2}
\]

2. Use the Product of Powers Rule to combine the exponents:
\[
x^6 \cdot x^{20} \cdot x^{-2} = x^{6+20-2} = x^{24}
\]
Therefore:
\[
6x^6 (x^5)^4 x^{-2} = 6x^{24}
\]

---

#### Expression 7:
\[
5x^{-1} (x^5)^2
\]

1. Simplify the power using the Power of a Power Rule:
\[
(x^5)^2 = x^{5 \cdot 2} = x^{10}
\]
So the expression becomes:
\[
5x^{-1} \cdot x^{10}
\]

2. Use the Product of Powers Rule to combine the exponents:
\[
x^{-1} \cdot x^{10} = x^{-1+10} = x^9
\]
Therefore:
\[
5x^{-1} (x^5)^2 = 5x^9
\]

---

#### Expression 8:
\[
\frac{2x^{-3} (x^{-3})^6}{9x^2 (x^{-3})^3}
\]

1. Simplify the powers using the Power of a Power Rule:
\[
(x^{-3})^6 = x^{-3 \cdot 6} = x^{-18}, \quad (x^{-3})^3 = x^{-3 \cdot 3} = x^{-9}
\]
So the expression becomes:
\[
\frac{2x^{-3} \cdot x^{-18}}{9x^2 \cdot x^{-9}}
\]

2. Use the Product of Powers Rule in the numerator and denominator:
\[
2x^{-3} \cdot x^{-18} = 2x^{-3-18} = 2x^{-21}, \quad 9x^2 \cdot x^{-9} = 9x^{2-9} = 9x^{-7}
\]
So the expression is now:
\[
\frac{2x^{-21}}{9x^{-7}}
\]

3. Simplify the coefficients and use the Quotient of Powers Rule:
\[
\frac{2}{9}, \quad \frac{x^{-21}}{x^{-7}} = x^{-21 - (-7)} = x^{-21 + 7} = x^{-14}
\]
Using the Negative Exponent Rule:
\[
x^{-14} = \frac{1}{x^{14}}
\]
Therefore:
\[
\frac{2x^{-3} (x^{-3})^6}{9x^2 (x^{-3})^3} = \frac{2}{9x^{14}}
\]

---

#### Expression 9:
\[
\frac{7x^3 (x^4)^6}{x^2 (x^{-3})^{-3}}
\]

1. Simplify the powers using the Power of a Power Rule:
\[
(x^4)^6 = x^{4 \cdot 6} = x^{24}, \quad (x^{-3})^{-3} = x^{-3 \cdot -3} = x^9
\]
So the expression becomes:
\[
\frac{7x^3 \cdot x^{24}}{x^2 \cdot x^9}
\]

2. Use the Product of Powers Rule in the numerator and denominator:
\[
7x^3 \cdot x^{24} = 7x^{3+24} = 7x^{27}, \quad x^2 \cdot x^9 = x^{2+9} = x^{11}
\]
So the expression is now:
\[
\frac{7x^{27}}{x^{11}}
\]

3. Simplify the coefficients and use the Quotient of Powers Rule:
\[
\frac{7}{1}, \quad \frac{x^{27}}{x^{11}} = x^{27-11} = x^{16}
\]
Therefore:
\[
\frac{7x^3 (x^4)^6}{x^2 (x^{-3})^{-3}} = 7x^{16}
\]

---

#### Expression 10:
\[
8x^{-5} (x^2)^3 x^{-3}
\]

1. Simplify the power using the Power of a Power Rule:
\[
(x^2)^3 = x^{2 \cdot 3} = x^6
\]
So the expression becomes:
\[
8x^{-5} \cdot x^6 \cdot x^{-3}
\]

2. Use the Product of Powers Rule to combine the exponents:
\[
x^{-5} \cdot x^6 \cdot x^{-3} = x^{-5+6-3} = x^{-2}
\]
Using the Negative Exponent Rule:
\[
x^{-2} = \frac{1}{x^2}
\]
Therefore:
\[
8x^{-5} (x^2)^3 x^{-3} = \frac{8}{x^2}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ x^{11} \\
2. & \ 4x^{11} \\
3. & \ \frac{2x^5}{7} \\
4. & \ 7x^{23} \\
5. & \ \frac{1}{x^{19}} \\
6. & \ 6x^{24} \\
7. & \ 5x^9 \\
8. & \ \frac{2}{9x^{14}} \\
9. & \ 7x^{16} \\
10. & \ \frac{8}{x^2}
\end{aligned}
}
\]