11. Usando propiedades, simplificar
a). x2x
7x
−3

b).
−3x
5y
2
2y−1x8 c).

x
5y−4
y−2
3

d). (2y
8
)(−5y
3
) e). (−4x
3
y
5
)(2x
6
y
−3
) f). (8x
6
y
−3
)
3

Let's simplify each expression using the properties of exponents. Remember the basic rules:

1. \(a^m \cdot a^n = a^{m+n}\) (Product of Powers Property)
2. \((a^m)^n = a^{m \cdot n}\) (Power of a Power Property)
3. \(\frac{a^m}{a^n} = a^{m-n}\) (Quotient of Powers Property)
4. \(a^{-n} = \frac{1}{a^n}\) (Negative Exponent Rule)

### a) \(x^2 \cdot x^7 \cdot x^{-3}\)

Combine the exponents by adding them, since they have the same base \(x\):

\[
x^2 \cdot x^7 \cdot x^{-3} = x^{2 + 7 - 3} = x^6
\]

### b) \(\frac{-3x^5y^2}{2y^{-1}x^8}\)

Simplify by applying the quotient of powers rule and then the negative exponent rule:

\[
\frac{-3x^5y^2}{2y^{-1}x^8} = -\frac{3}{2} \cdot x^{5 - 8} \cdot y^{2 - (-1)}
\]

\[
= -\frac{3}{2} \cdot x^{-3} \cdot y^{3}
\]

Convert the negative exponent to a positive by taking the reciprocal:

\[
= -\frac{3y^3}{2x^3}
\]

### c) \(\frac{x^5y^{-4}}{y^{-2}}\)

Simplify by applying the quotient of powers rule:

\[
\frac{x^5y^{-4}}{y^{-2}} = x^5 \cdot y^{-4 - (-2)}
\]

\[
= x^5 \cdot y^{-4 + 2} = x^5 \cdot y^{-2}
\]

Convert the negative exponent to a positive by taking the reciprocal:

\[
= \frac{x^5}{y^2}
\]

### d) \((2y^8)(-5y^3)\)

Combine the exponents by multiplying the coefficients and adding the exponents of \(y\):

\[
(2y^8)(-5y^3) = (2 \cdot -5) \cdot y^{8 + 3}
\]

\[
= -10y^{11}
\]

### e) \((-4x^3y^5)(2x^6y^{-3})\)

Multiply the coefficients and use the product of powers property:

\[
(-4x^3y^5)(2x^6y^{-3}) = (-4 \cdot 2) \cdot x^{3+6} \cdot y^{5 + (-3)}
\]

\[
= -8x^9y^{2}
\]

### f) \((8x^6y^{-3})^3\)

Apply the power of a power property to each factor inside the parentheses:

\[
(8x^6y^{-3})^3 = 8^3 \cdot (x^6)^3 \cdot (y^{-3})^3
\]

Calculate each exponent:

\[
8^3 = 512, \quad (x^6)^3 = x^{18}, \quad (y^{-3})^3 = y^{-9}
\]

So, the expression simplifies to:

\[
512x^{18}y^{-9}
\]

Convert the negative exponent to a positive by taking the reciprocal:

\[
= \frac{512x^{18}}{y^9}
\]

### Summary of Simplified Expressions

a) \(x^6\)

b) \(-\frac{3y^3}{2x^3}\)

c) \(\frac{x^5}{y^2}\)

d) \(-10y^{11}\)

e) \(-8x^9y^2\)

f) \(\frac{512x^{18}}{y^9}\)

Would you like more details on any part, or do you have any other questions?

### Related Questions:

1. How would you simplify \((x^2y^{-3})^4\)?
2. What is the result of \(\frac{a^5b^3}{a^2b^{-2}}\)?
3. How do you simplify \((3x^{-2}y^4)^2\)?
4. What is \(\frac{(4x^3y^{-2})^2}{(2x^{-1}y)^3}\)?
5. How would you simplify \(5x^0 \cdot 4y^{-1}\)?

### Tip:

When simplifying expressions with exponents, carefully follow the rules for multiplying, dividing, and raising powers to ensure accuracy.

Learn how to simplify exponential expressions step-by-step with detailed solutions. This page covers key concepts such as product of powers, quotient of powers, power of a power, and negative exponent rules. Suitable for high school students.

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