I Love Exponents Worksheet - Free Printable

Problem Analysis:

The task involves simplifying a complex algebraic expression. The given expression is:

\[
\frac{\left(3x^{-6}y^2\right)^3 \cdot 2x^{10}y^{-7}}{(2xy)^2 \cdot 10x^3y^{11}} \div \frac{54x^8y^{-1}}{8x^{10}y^4}
\]

We will break this down step by step.

---

Step 1: Simplify the numerator of the first fraction

The numerator is:

\[
\left(3x^{-6}y^2\right)^3 \cdot 2x^{10}y^{-7}
\]

#### Step 1.1: Simplify \(\left(3x^{-6}y^2\right)^3\)
Using the power rule \((ab)^n = a^n b^n\) and \((x^m)^n = x^{mn}\):

\[
\left(3x^{-6}y^2\right)^3 = 3^3 \cdot (x^{-6})^3 \cdot (y^2)^3 = 27 \cdot x^{-18} \cdot y^6
\]

So, the expression becomes:

\[
27x^{-18}y^6 \cdot 2x^{10}y^{-7}
\]

#### Step 1.2: Multiply the terms
Using the rules of exponents \(a^m \cdot a^n = a^{m+n}\):

\[
27x^{-18}y^6 \cdot 2x^{10}y^{-7} = (27 \cdot 2) \cdot x^{-18+10} \cdot y^{6-7} = 54x^{-8}y^{-1}
\]

Thus, the simplified numerator is:

\[
54x^{-8}y^{-1}
\]

---

Step 2: Simplify the denominator of the first fraction

The denominator is:

\[
(2xy)^2 \cdot 10x^3y^{11}
\]

#### Step 2.1: Simplify \((2xy)^2\)
Using the power rule \((ab)^n = a^n b^n\):

\[
(2xy)^2 = 2^2 \cdot x^2 \cdot y^2 = 4x^2y^2
\]

So, the expression becomes:

\[
4x^2y^2 \cdot 10x^3y^{11}
\]

#### Step 2.2: Multiply the terms
Using the rules of exponents \(a^m \cdot a^n = a^{m+n}\):

\[
4x^2y^2 \cdot 10x^3y^{11} = (4 \cdot 10) \cdot x^{2+3} \cdot y^{2+11} = 40x^5y^{13}
\]

Thus, the simplified denominator is:

\[
40x^5y^{13}
\]

---

Step 3: Simplify the first fraction

The first fraction is:

\[
\frac{54x^{-8}y^{-1}}{40x^5y^{13}}
\]

Using the rule \(\frac{a^m}{a^n} = a^{m-n}\):

\[
\frac{54x^{-8}y^{-1}}{40x^5y^{13}} = \frac{54}{40} \cdot x^{-8-5} \cdot y^{-1-13} = \frac{54}{40} \cdot x^{-13} \cdot y^{-14}
\]

Simplify the coefficient \(\frac{54}{40}\):

\[
\frac{54}{40} = \frac{27}{20}
\]

So, the simplified first fraction is:

\[
\frac{27}{20}x^{-13}y^{-14}
\]

---

Step 4: Simplify the second fraction

The second fraction is:

\[
\frac{54x^8y^{-1}}{8x^{10}y^4}
\]

Using the rule \(\frac{a^m}{a^n} = a^{m-n}\):

\[
\frac{54x^8y^{-1}}{8x^{10}y^4} = \frac{54}{8} \cdot x^{8-10} \cdot y^{-1-4} = \frac{54}{8} \cdot x^{-2} \cdot y^{-5}
\]

Simplify the coefficient \(\frac{54}{8}\):

\[
\frac{54}{8} = \frac{27}{4}
\]

So, the simplified second fraction is:

\[
\frac{27}{4}x^{-2}y^{-5}
\]

---

Step 5: Perform the division of the two fractions

The original expression is:

\[
\frac{\frac{27}{20}x^{-13}y^{-14}}{\frac{27}{4}x^{-2}y^{-5}}
\]

Dividing by a fraction is equivalent to multiplying by its reciprocal:

\[
\frac{\frac{27}{20}x^{-13}y^{-14}}{\frac{27}{4}x^{-2}y^{-5}} = \frac{27}{20}x^{-13}y^{-14} \cdot \frac{4}{27}x^2y^5
\]

#### Step 5.1: Simplify the coefficients
\[
\frac{27}{20} \cdot \frac{4}{27} = \frac{27 \cdot 4}{20 \cdot 27} = \frac{4}{20} = \frac{1}{5}
\]

#### Step 5.2: Simplify the variables
Using the rule \(a^m \cdot a^n = a^{m+n}\):

\[
x^{-13} \cdot x^2 = x^{-13+2} = x^{-11}
\]
\[
y^{-14} \cdot y^5 = y^{-14+5} = y^{-9}
\]

So, the expression becomes:

\[
\frac{1}{5}x^{-11}y^{-9}
\]

---

Final Answer:

The simplified expression is:

\[
\boxed{\frac{1}{5x^{11}y^9}}
\]