v\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)
4^0 ⋅ 5^−2/(−2)^−3
To apply the properties of integer exponents, we can rewrite the expression as follows:
4^0 ⋅ 5^(-2) / (-2)^(-3)
= 1 ⋅ 1 / ((-1)^3/2)^(-3)
= 1 / ((-1)^(3/2))^(-3)
Now, let's simplify the exponents:
(-1)^(3/2) = -1 * √(-1) = -1 * i = -i (since the square root of -1 is defined as i)
Now we have:
1 / (-i)^(-3)
= 1 / [(1/(-i))^3]
= 1 / (1/(-i^3))
= 1 / (1/i)
= 1 / (1/i) * (i/i)
= 1 / (i^2)
= 1 / (-1)
= -1
Therefore, the expression simplifies to -1.
To apply the properties of integer exponents, we can use the following rules:
1. Any number raised to the power of 0 is equal to 1.
2. A negative exponent can be changed to a positive exponent by taking the reciprocal of the base.
Using these rules, let's rewrite the expression step-by-step:
1. Start with the given expression: 4^0 ⋅ 5^−2/(−2)^−3
2. Since 4^0 is equal to 1, we can replace it: 1 ⋅ 5^−2/(−2)^−3
3. Now, let's deal with the negative exponents. Since 5^−2 is on the numerator, we'll keep it as it is. However, for the denominator (−2)^−3, we'll change it to a positive exponent by taking the reciprocal of the base: 1 ⋅ 5^−2/((−1/2)^3)
4. Simplifying further, we have: 5^−2/(−1/2)^3
5. Let's simplify the reciprocal of the denominator: 5^−2/((−1/2)^3) = 5^−2/((-1)^3/(2^3)) = 5^−2/(−1/8)
6. Now, let's deal with the negative exponent. We can rewrite 5^−2 as (1/5^2): (1/5^2)/(−1/8)
7. Simplifying the expression, we get: (1/25)/(−1/8)
To divide fractions, we multiply the numerator by the reciprocal (flip) of the denominator:
(1/25)/(−1/8) = (1/25) * (−8/1)
Finally, multiplying the numerators and denominators, we obtain:
(-8/25)
Therefore, the simplified fraction is -8/25.