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)
Using the properties of integer exponents, we can rewrite 4^0 ⋅ 5^(-2)/(-2)^(-3) as (1 ⋅ 1)/(5^2/2^3).
Simplifying further, we have 1/[(25/4)^(1/3)].
Now, we can simplify the expression inside the brackets to obtain: 1/[∛(25/4)].
To simplify this further, we can express the fraction 25/4 as an improper fraction: 25/4 = 6 1/4.
Therefore, the expression becomes 1/[∛(6 1/4)].
Taking the cube root of 6 1/4, we obtain: 1/∛(25/4) = 1/(2/∛(1)) = 1/(2/1) = 1/2.
So, the final answer is 1/2.
To apply the properties of integer exponents and generate equivalent expressions, we can follow these rules:
1. Any number raised to the power of 0 is equal to 1.
2. To convert a negative exponent to a positive exponent, we can take the reciprocal of the base and make the exponent positive.
3. When dividing with the same base, we subtract the exponents.
Let's apply these rules to the given expression step by step:
First, we'll address the exponents in 4^0 ⋅ 5^−2:
1. 4^0 = 1 (any number raised to the power of 0 is 1).
2. 5^−2 = 1/5^2 (taking reciprocal to change the negative exponent to a positive exponent).
After applying the above rules, the expression becomes:
1 ⋅ 1/(5^2) / (−2)−3
Next, we'll simplify (−2)−3:
1. (-2)^-3 = 1/(-2)^3 (taking reciprocal to change the negative exponent to a positive exponent).
The expression now becomes:
1 ⋅ 1/(5^2) / 1/(-2)^3
Further simplifying, we have:
1 ⋅ 1/(25) / 1/(-8)
To simplify this expression, we can multiply the numerator and the denominator of the second fraction by (-8):
1 ⋅ 1/(25) / 1/(-8) * (-8)/(-8)
This simplifies to:
1 ⋅ 1/(25) / (-8)/(-8)
Which simplifies to:
1/25 * (-8)/(-8)
Now we can multiply the numerators and the denominators:
-8 / (25 * -8)
Simplifying further, we have:
-8 / -200
Finally, we simplify the fraction:
-8/200 = -1/25
Therefore, the result of the expression 4^0 ⋅ 5^−2/(−2)−3, using the properties of integer exponents and simplifying, is -1/25.