I Love Exponents Worksheet - Free Printable

It looks like you've uploaded an image containing a math problem involving algebraic expressions with exponents. Let's carefully analyze and solve the problem step by step.

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Problem:

We are given the following expression:

$$
\frac{(3x^{-6}y^2)^3 \cdot 2x^{10}y^{-7}}{(2xy)^2 \cdot 10x^3y^{11}} \div \frac{54}{x^8}
$$

Our goal is to simplify this expression completely.

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Step 1: Simplify the numerator

The numerator is:
$$
(3x^{-6}y^2)^3 \cdot 2x^{10}y^{-7}
$$

First, expand $(3x^{-6}y^2)^3$:
- $3^3 = 27$
- $(x^{-6})^3 = x^{-18}$
- $(y^2)^3 = y^6$

So:
$$
(3x^{-6}y^2)^3 = 27x^{-18}y^6
$$

Now multiply by $2x^{10}y^{-7}$:
$$
27x^{-18}y^6 \cdot 2x^{10}y^{-7} = (27 \cdot 2)x^{-18+10}y^{6-7} = 54x^{-8}y^{-1}
$$

So, the numerator simplifies to:
$$
54x^{-8}y^{-1}
$$

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Step 2: Simplify the denominator

The denominator is:
$$
(2xy)^2 \cdot 10x^3y^{11}
$$

First, expand $(2xy)^2$:
- $2^2 = 4$
- $x^2$, $y^2$

So:
$$
(2xy)^2 = 4x^2y^2
$$

Now multiply by $10x^3y^{11}$:
$$
4x^2y^2 \cdot 10x^3y^{11} = (4 \cdot 10)x^{2+3}y^{2+11} = 40x^5y^{13}
$$

So, the denominator simplifies to:
$$
40x^5y^{13}
$$

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Step 3: Put it all together

Now we have:
$$
\frac{54x^{-8}y^{-1}}{40x^5y^{13}} \div \frac{54}{x^8}
$$

Wait — actually, the original expression is:
$$
\frac{\text{numerator}}{\text{denominator}} \div \frac{54}{x^8}
$$

So first compute:
$$
\frac{54x^{-8}y^{-1}}{40x^5y^{13}} = \frac{54}{40} \cdot x^{-8-5} \cdot y^{-1-13} = \frac{27}{20} x^{-13} y^{-14}
$$

Now divide this by $\frac{54}{x^8}$:
$$
\left(\frac{27}{20} x^{-13} y^{-14}\right) \div \left(\frac{54}{x^8}\right)
$$

Dividing by a fraction is multiplying by its reciprocal:
$$
= \frac{27}{20} x^{-13} y^{-14} \cdot \frac{x^8}{54}
$$

Now simplify:
- $\frac{27}{54} = \frac{1}{2}$
- So: $\frac{1}{2} \cdot \frac{1}{20} = \frac{1}{40}$
- $x^{-13} \cdot x^8 = x^{-5}$
- $y^{-14}$ remains

So the result is:
$$
\frac{1}{40} x^{-5} y^{-14}
$$

We can write this as:
$$
\frac{1}{40x^5y^{14}}
$$

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✔ Final Answer:

$$
\boxed{\frac{1}{40x^5y^{14}}}
$$

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Let me know if you'd like a step-by-step breakdown of any part!