10. Calcular (usando propiedades)
a). 2
−3

b).
3
4
3

c).
−5
3
−2

d). − 7
2

To solve these problems using the properties of exponents, we'll evaluate each expression step by step.

### a) \(2^{-3}\)

To solve this, recall the property of exponents that states \(a^{-n} = \frac{1}{a^n}\).

\[
2^{-3} = \frac{1}{2^3} = \frac{1}{8}
\]

### b) \(3^4 \cdot 3^3\)

Here, we use the property that says when multiplying two exponents with the same base, you add the exponents: \(a^m \cdot a^n = a^{m+n}\).

\[
3^4 \cdot 3^3 = 3^{4+3} = 3^7
\]

We can calculate \(3^7\) if needed:

\[
3^7 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 2187
\]

So, \(3^4 \cdot 3^3 = 2187\).

### c) \((-5)^3 \cdot (-5)^{-2}\)

For this one, we use the same property as in part (b) and also apply the negative exponent rule. 

First, combine the exponents:

\[
(-5)^3 \cdot (-5)^{-2} = (-5)^{3 + (-2)} = (-5)^{1} = -5
\]

So, \((-5)^3 \cdot (-5)^{-2} = -5\).

### d) \(-7^2\)

It's important to notice that the negative sign is not part of the exponentiation when it's outside like this:

\[
-7^2 = -(7^2)
\]

So, compute \(7^2\):

\[
7^2 = 49
\]

Therefore:

\[
-7^2 = -49
\]

### Summary of Solutions

a) \(2^{-3} = \frac{1}{8}\)

b) \(3^4 \cdot 3^3 = 2187\)

c) \((-5)^3 \cdot (-5)^{-2} = -5\)

d) \(-7^2 = -49\)

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

### Related Questions:

1. What is the result of \(4^{-2}\)?
2. How would you simplify \(2^3 \cdot 2^{-5}\)?
3. How do you evaluate \( (10^{-1})^3 \)?
4. What is \(5^0\) and why?
5. How do you simplify \( x^a \cdot x^{-b} \)?

### Tip:

When dealing with negative exponents, remember that \(a^{-n} = \frac{1}{a^n}\). This can help simplify expressions and convert negative exponents into positive ones.

Explore how to solve exponential expressions involving negative and positive bases. Learn the rules of exponents, including negative exponent rules and multiplication of exponents. Suitable for high school students.

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