Apply the Properties of Integer Exponents to generate equivalent expressions to 4^0 ⋅ 5^−/2(−2)−3 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
To apply the properties of integer exponents, we can use the following rules:
1. $a^0 = 1$ for any non-zero number $a$.
2. $a^{-n} = \frac{1}{a^n}$ for any non-zero number $a$.
3. $a^m \cdot a^n = a^{m+n}$ for any non-zero number $a$.
Using these properties, we can simplify the expression.
First, we apply rule 1 to $4^0$ and rule 2 to $5^{-2}$. This gives us:
$4^0 \cdot 5^{-2}(-2)^{-3}$
$= 1 \cdot \frac{1}{5^2} \cdot \frac{1}{(-2)^3}$
Next, we use rule 3 to combine the exponents of 5 and -2:
$= \frac{1}{5^2 \cdot (-2)^3}$
To simplify further, we can expand the exponents:
$= \frac{1}{5^2 \cdot (-2)^3}$
$= \frac{1}{25 \cdot (-8)}$
$= \frac{1}{-200}$
$= -\frac{1}{200}$
Therefore, the simplified fraction with no remaining exponents is $-\frac{1}{200}$.
To generate equivalent expressions with only positive exponents, we can use the properties of integer exponents.
First, let's rewrite the expression using the property of negative exponents:
4^0 ⋅ 5^(-1/2) ⋅ (-2)^(-3)
Next, we can use the property of zero exponents to simplify 4^0:
1 ⋅ 5^(-1/2) ⋅ (-2)^(-3)
Now, let's simplify the expression further by moving the negative exponents to the denominator using the property a^(-n) = 1/a^n:
1/5^(1/2) ⋅ 1/(-2)^3
Simplifying the exponents and the negative sign:
1/√5 ⋅ 1/(-8)
Finally, we can simplify the expression by multiplying the numerators and denominators:
1/√5 ⋅ -1/8 = -1/(8√5)
Therefore, the simplified fraction with no remaining exponents is -1/(8√5).