apply the properties of integer exponents to generate equivalent expressions to 4^0 times 5^-2\ (-2)^-3 with only positive exponents,then solve the expression. Your answer will be a simplified fraction with no remaining exponents
To generate equivalent expressions with only positive exponents, we can use the following properties of integer exponents:
1. Any number raised to the power of 0 is equal to 1: a^0 = 1
2. When dividing two numbers with the same base, we subtract the exponents: a^m / a^n = a^(m-n)
3. When raising a power to another power, we multiply the exponents: (a^m)^n = a^(m*n)
4. Negative exponents can be converted to positive exponents by taking the reciprocal of the base: a^(-n) = 1 / a^n
Let's apply these properties step by step:
4^0 * 5^(-2) * (-2)^(-3)
Step 1: Apply property 1 to 4^0 and 5^(-2):
1 * 1/5^2 * (-2)^(-3)
Step 2: Simplify the negative exponents using property 4:
1 * 1/(5^2) * (1/(-2)^3)
Step 3: Simplify the base with a negative exponent using property 4:
1 * 1/(5^2) * (1/(-8))
Step 4: Simplify the remaining expression:
1 * 1/25 * 1/(-8) = -1/(25*8) = -1/200
Thus, the simplified fraction of the expression 4^0 * 5^-2 * (-2)^-3 with no remaining exponents is -1/200.