simplifying exponent expressions Math Worksheets, Math Practice ... - Free Printable

To solve the given problems involving simplifying exponent expressions, we will use the following properties of exponents:

1. Power of a Power: \((x^a)^b = x^{a \cdot b}\)
2. Product of Powers: \(x^a \cdot x^b = x^{a + b}\)
3. Quotient of Powers: \(\frac{x^a}{x^b} = x^{a - b}\)
4. Negative Exponent: \(x^{-a} = \frac{1}{x^a}\)

Let's solve each problem step by step.

---

Problem 1:

\[
\frac{3x^8 (x^2)^6}{3x^5 (x^2)^2}
\]

#### Step 1: Simplify the numerator and denominator using the power of a power rule.
- Numerator: \(3x^8 (x^2)^6 = 3x^8 \cdot x^{2 \cdot 6} = 3x^8 \cdot x^{12}\)
- Denominator: \(3x^5 (x^2)^2 = 3x^5 \cdot x^{2 \cdot 2} = 3x^5 \cdot x^4\)

So the expression becomes:
\[
\frac{3x^8 \cdot x^{12}}{3x^5 \cdot x^4}
\]

#### Step 2: Combine the powers of \(x\) using the product of powers rule.
- Numerator: \(x^8 \cdot x^{12} = x^{8 + 12} = x^{20}\)
- Denominator: \(x^5 \cdot x^4 = x^{5 + 4} = x^9\)

The expression now is:
\[
\frac{3x^{20}}{3x^9}
\]

#### Step 3: Simplify the coefficients and the powers of \(x\) using the quotient of powers rule.
- Coefficients: \(\frac{3}{3} = 1\)
- Powers of \(x\): \(\frac{x^{20}}{x^9} = x^{20 - 9} = x^{11}\)

So the simplified expression is:
\[
x^{11}
\]

---

Problem 2:

\[
4x^{-4} (x^4)^4 x^{-1}
\]

#### Step 1: Simplify \((x^4)^4\) using the power of a power rule.
\[
(x^4)^4 = x^{4 \cdot 4} = x^{16}
\]

So the expression becomes:
\[
4x^{-4} \cdot x^{16} \cdot x^{-1}
\]

#### Step 2: Combine all the powers of \(x\) using the product of powers rule.
\[
x^{-4} \cdot x^{16} \cdot x^{-1} = x^{-4 + 16 - 1} = x^{11}
\]

So the expression is:
\[
4x^{11}
\]

---

Problem 3:

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

#### Step 1: Simplify \((x^2)^4\) and \((x^{-3})^{-3}\) using the power of a power rule.
- \((x^2)^4 = x^{2 \cdot 4} = x^8\)
- \((x^{-3})^{-3} = x^{-3 \cdot -3} = x^9\)

So the expression becomes:
\[
\frac{2x^3 \cdot x^8}{7x^{-3} \cdot x^9}
\]

#### Step 2: Combine the powers of \(x\) in the numerator and denominator using the product of powers rule.
- Numerator: \(x^3 \cdot x^8 = x^{3 + 8} = x^{11}\)
- Denominator: \(x^{-3} \cdot x^9 = x^{-3 + 9} = x^6\)

The expression now is:
\[
\frac{2x^{11}}{7x^6}
\]

#### Step 3: Simplify the powers of \(x\) using the quotient of powers rule.
\[
\frac{x^{11}}{x^6} = x^{11 - 6} = x^5
\]

So the expression is:
\[
\frac{2x^5}{7}
\]

---

Problem 4:

\[
7x^{-2} (x^5)^5
\]

#### Step 1: Simplify \((x^5)^5\) using the power of a power rule.
\[
(x^5)^5 = x^{5 \cdot 5} = x^{25}
\]

So the expression becomes:
\[
7x^{-2} \cdot x^{25}
\]

#### Step 2: Combine the powers of \(x\) using the product of powers rule.
\[
x^{-2} \cdot x^{25} = x^{-2 + 25} = x^{23}
\]

So the expression is:
\[
7x^{23}
\]

---

Problem 5:

\[
\frac{x^4 (x^6)^{-2}}{x^3 (x^4)^2}
\]

#### Step 1: Simplify \((x^6)^{-2}\) and \((x^4)^2\) using the power of a power rule.
- \((x^6)^{-2} = x^{6 \cdot -2} = x^{-12}\)
- \((x^4)^2 = x^{4 \cdot 2} = x^8\)

So the expression becomes:
\[
\frac{x^4 \cdot x^{-12}}{x^3 \cdot x^8}
\]

#### Step 2: Combine the powers of \(x\) in the numerator and denominator using the product of powers rule.
- Numerator: \(x^4 \cdot x^{-12} = x^{4 - 12} = x^{-8}\)
- Denominator: \(x^3 \cdot x^8 = x^{3 + 8} = x^{11}\)

The expression now is:
\[
\frac{x^{-8}}{x^{11}}
\]

#### Step 3: Simplify the powers of \(x\) using the quotient of powers rule.
\[
\frac{x^{-8}}{x^{11}} = x^{-8 - 11} = x^{-19}
\]

Using the negative exponent rule:
\[
x^{-19} = \frac{1}{x^{19}}
\]

So the expression is:
\[
\frac{1}{x^{19}}
\]

---

Problem 6:

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

#### Step 1: Simplify \((x^5)^4\) using the power of a power rule.
\[
(x^5)^4 = x^{5 \cdot 4} = x^{20}
\]

So the expression becomes:
\[
6x^6 \cdot x^{20} \cdot x^{-2}
\]

#### Step 2: Combine all the powers of \(x\) using the product of powers rule.
\[
x^6 \cdot x^{20} \cdot x^{-2} = x^{6 + 20 - 2} = x^{24}
\]

So the expression is:
\[
6x^{24}
\]

---

Problem 7:

\[
5x^{-1} (x^5)^2
\]

#### Step 1: Simplify \((x^5)^2\) using the power of a power rule.
\[
(x^5)^2 = x^{5 \cdot 2} = x^{10}
\]

So the expression becomes:
\[
5x^{-1} \cdot x^{10}
\]

#### Step 2: Combine the powers of \(x\) using the product of powers rule.
\[
x^{-1} \cdot x^{10} = x^{-1 + 10} = x^9
\]

So the expression is:
\[
5x^9
\]

---

Problem 8:

\[
\frac{2x^{-3} (x^{-3})^6}{9x^2 (x^{-3})^3}
\]

#### Step 1: Simplify \((x^{-3})^6\) and \((x^{-3})^3\) using the power of a power rule.
- \((x^{-3})^6 = x^{-3 \cdot 6} = x^{-18}\)
- \((x^{-3})^3 = x^{-3 \cdot 3} = x^{-9}\)

So the expression becomes:
\[
\frac{2x^{-3} \cdot x^{-18}}{9x^2 \cdot x^{-9}}
\]

#### Step 2: Combine the powers of \(x\) in the numerator and denominator using the product of powers rule.
- Numerator: \(x^{-3} \cdot x^{-18} = x^{-3 - 18} = x^{-21}\)
- Denominator: \(x^2 \cdot x^{-9} = x^{2 - 9} = x^{-7}\)

The expression now is:
\[
\frac{2x^{-21}}{9x^{-7}}
\]

#### Step 3: Simplify the powers of \(x\) using the quotient of powers rule.
\[
\frac{x^{-21}}{x^{-7}} = x^{-21 - (-7)} = x^{-21 + 7} = x^{-14}
\]

Using the negative exponent rule:
\[
x^{-14} = \frac{1}{x^{14}}
\]

So the expression is:
\[
\frac{2}{9x^{14}}
\]

---

Problem 9:

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

#### Step 1: Simplify \((x^4)^6\) and \((x^{-3})^{-3}\) using the power of a power rule.
- \((x^4)^6 = x^{4 \cdot 6} = x^{24}\)
- \((x^{-3})^{-3} = x^{-3 \cdot -3} = x^9\)

So the expression becomes:
\[
\frac{7x^3 \cdot x^{24}}{x^2 \cdot x^9}
\]

#### Step 2: Combine the powers of \(x\) in the numerator and denominator using the product of powers rule.
- Numerator: \(x^3 \cdot x^{24} = x^{3 + 24} = x^{27}\)
- Denominator: \(x^2 \cdot x^9 = x^{2 + 9} = x^{11}\)

The expression now is:
\[
\frac{7x^{27}}{x^{11}}
\]

#### Step 3: Simplify the powers of \(x\) using the quotient of powers rule.
\[
\frac{x^{27}}{x^{11}} = x^{27 - 11} = x^{16}
\]

So the expression is:
\[
7x^{16}
\]

---

Problem 10:

\[
8x^{-5} (x^2)^3 x^{-3}
\]

#### Step 1: Simplify \((x^2)^3\) using the power of a power rule.
\[
(x^2)^3 = x^{2 \cdot 3} = x^6
\]

So the expression becomes:
\[
8x^{-5} \cdot x^6 \cdot x^{-3}
\]

#### Step 2: Combine all the powers of \(x\) using the product of powers rule.
\[
x^{-5} \cdot x^6 \cdot x^{-3} = x^{-5 + 6 - 3} = x^{-2}
\]

Using the negative exponent rule:
\[
x^{-2} = \frac{1}{x^2}
\]

So the expression is:
\[
\frac{8}{x^2}
\]

---

Final Answers:

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