Apply the Properties of Integer Exponents to generate equivalent expressions to 40 ⋅ 5−2(−2)−3 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
To apply the properties of integer exponents and generate equivalent expressions with only positive exponents for 40 ⋅ 5^(-2)(-2)^(-3), we can start by using the rule that (a^m)^n = a^(m * n).
We have 40 ⋅ 5^(-2)(-2)^(-3). First, let's simplify the exponents:
5^(-2) = 1/5^2 = 1/25
(-2)^(-3) = 1/(-2)^3 = 1/(-8) = -1/8
Now, substitute these values back into the original expression:
40 ⋅ 1/25 ⋅ -1/8
Next, we can use the property of multiplication to simplify the expression:
40 * 1/25 * -1/8 = (40 * -1 * 1) / (25 * 8) = -40 / 200 = -1/5
Hence, the simplified fraction with no remaining exponents for the expression 40 ⋅ 5^(-2)(-2)^(-3) is -1/5.
To apply the properties of integer exponents, let's break down the expression step by step:
40 ⋅ 5−2(−2)−3
First, let's simplify the negative exponents:
40 ⋅ 5^(-2) ⋅ (-2)^(-2) ⋅ (-2)^(-3)
Now, let's apply the property of a negative exponent:
40 ⋅ (1/5^2) ⋅ (1/(-2)^2) ⋅ (1/(-2)^3)
Next, let's simplify the exponents:
40 ⋅ (1/25) ⋅ (1/4) ⋅ (1/(-8))
Now, let's multiply the numerators and denominators:
40 ⋅ 1 ⋅ 1 ⋅ 1 / (25 ⋅ 4 ⋅ (-8))
Simplifying further:
40 / (100 ⋅ (-8))
Now, let's multiply the numbers in the denominator:
40 / (-800)
Finally, let's simplify the fraction:
40 / (-800) = -1/20
Therefore, the expression 40 ⋅ 5^(-2)(-2)^(-2)(-2)^(-3) simplifies to -1/20.