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)
Recall the properties of integer exponents:
1) a^m * a^n = a^(m + n)
2) a^m / a^n = a^(m - n)
3) (a^m)^n = a^(m * n)
4) a^(-n) = 1 / a^n
Using these properties, we can simplify the given expression:
4^0 * 5^(-2) * (-2)^(-3)
Since any number raised to the power of 0 is equal to 1, we can simplify:
1 * 5^(-2) * (-2)^(-3)
Applying property 4), we can rewrite the expression as:
1 / 5^2 * 1 / (-2)^3
Simplifying further:
1 / 25 * 1 / (-2^3)
= 1 / 25 * 1 / (-8)
= 1 / (-200)
The simplified fraction with no remaining exponents is -1/200.
To generate equivalent expressions with only positive exponents, let's apply the properties of integer exponents.
Property 1: x^0 = 1 for any non-zero value of x.
We can apply this property to the base 4, so we have 4^0 = 1.
Property 2: x^(-n) = 1 / (x^n) for any non-zero value of x.
We can apply this property to the bases 5 and -2.
5^(-2) = 1 / (5^2) = 1 / 25
(-2)^(-3) = 1 / ((-2)^3) = 1 / (-8) = -1/8
Now let's rewrite the expression using these properties:
4^0 * 5^(-2) * (-2)^(-3)
= 1 * (1/25) * (-1/8)
= -1 / (25 * 8)
= -1 / 200
Therefore, the simplified fraction for the expression 4^0 * 5^(-2) * (-2)^(-3) is -1/200.