Question 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.(1 point) 40 ⋅ 5−2(−2)−3=
To apply the properties of integer exponents and generate equivalent expressions with only positive exponents, we can use the following rules:
1. For any non-zero number x, x^(-n) = 1/x^n.
2. For any non-zero number x, x^0 = 1.
3. For any non-zero numbers a and b, a^n * b^n = (a * b)^n.
Using these rules, let's simplify the expression:
40 ⋅ 5^(-2)(-2)^(-3)
Using rule 1 for 5^(-2), we have:
40 ⋅ (1/5^2)(-2)^(-3)
Simplifying 5^2 = 25:
40 ⋅ (1/25)(-2)^(-3)
Using rule 2 for (-2)^(-3), we have:
40 ⋅ (1/25)(1/(-2)^3)
Simplifying (-2)^3 = -8:
40 ⋅ (1/25)(1/(-8))
Simplifying 1/(-8):
40 ⋅ (1/25)(-1/8)
Using rule 3 for (1/25)(-1/8), we have:
(40 * 1 * -1) / (25 * 8) = -40 / 200
Simplifying the fraction:
-40 / 200 = -1 / 5
Therefore, the solution is -1 / 5.
To generate equivalent expressions with positive exponents, we can apply the following properties of integer exponents:
1. Product of Powers Property: a^m * a^n = a^(m+n)
2. Quotient of Powers Property: a^m / a^n = a^(m-n)
3. Power of a Power Property: (a^m)^n = a^(m*n)
4. Negative Exponent Property: a^(-n) = 1 / a^n
Now let's apply these properties to the given expression:
40 * 5^(-2) * (-2)^(-2) * (-2)^(-3)
Using the negative exponent property, we can write the expression as:
40 * 1/5^2 * 1/(-2)^2 * 1/(-2)^3
Simplifying the exponents:
40 * 1/25 * 1/4 * 1/(-8)
Combining the fractions:
= (40 * 1 * 1 * 1) / (25 * 4 * (-8))
Simplifying the numerator:
= 40 / (25 * 4 * (-8))
Simplifying the denominator:
= 40 / (-800)
Dividing:
= -1/20
Therefore, the simplified fraction is -1/20.
To generate equivalent expressions with only positive exponents, we can use the properties of integer exponents.
First, let's start by simplifying the expression without worrying about the exponents:
40 ⋅ 5^(-2) * (-2)^(-3)
Next, we can apply the properties of integer exponents. The properties are as follows:
1. x^(-n) = 1/x^n
2. (x^m)^n = x^(m * n)
3. x^0 = 1
Using property 1, we can rewrite 5^(-2) as 1/5^2 and (-2)^(-3) as 1/(-2)^3.
40 ⋅ 1/(5^2) * 1/(-2)^3
Simplifying further, we can calculate the exponents:
40 ⋅ 1/25 * 1/(-8)
Now, we can simplify the expression by multiplying and dividing:
(40 * 1 * 1) / (25 * (-8))
40 / (-200)
Finally, we can simplify the fraction. Since both 40 and 200 are divisible by 40, we can divide each by 40:
1 / (-5)
Therefore, the final answer is -1/5.