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, we need to work step-by-step through the expression.
Step 1: Deal with the negative exponents first. In this case, we have 5^(-2) and 5^(-3). The property states that a number raised to a negative exponent is equal to 1 over that number raised to the positive exponent.
So, 5^(-2) = 1/5^2 and 5^(-3) = 1/5^3.
Step 2: Simplify any exponent expressions. In this case, we have (-2)(-2) which equals 4.
So now the expression becomes:
40 * (1/5^2)(4)(1/5^3)
Step 3: Simplify the numerical expression. 40 * 4 is equal to 160.
So the expression becomes:
160 * (1/5^2)(1/5^3)
Step 4: Simplify the exponent expression. When multiplying exponent expressions with the same base, we add the exponents.
So 5^2 * 5^3 = 5^(2+3) = 5^5.
Now the expression becomes:
160 * (1/5^5).
Step 5: Simplify the expression further. 1/5^5 means 1 divided by 5^5.
So the final expression is:
160/5^5.
This can also be written as a simplified fraction:
32/3125.
Therefore, the equivalent expression to 40 ⋅ 5^(-2)(-2)^(-3) with positive exponents is 32/3125.
To apply the properties of integer exponents, we need to follow the following rules:
1. a^m ⋅ a^n = a^(m+n)
2. a^(-n) = 1/a^n
Let's apply these rules step-by-step to generate equivalent expressions:
Step 1: Rewrite the expression with positive exponents:
40 ⋅ 5^(-2)(-2)^(-3)
Step 2: Simplify the negative exponents using Rule 2:
40 ⋅ 1/(5^2)(1/(-2)^3)
Step 3: Simplify the values of the exponents:
40 ⋅ 1/(25)(1/(-8))
Step 4: Evaluate the multiplication and the division:
40 ⋅ 1/25 ⋅ (-8)
Step 5: Simplify the expression:
-320/25 or -64/5
Therefore, the simplified fraction of the expression is -64/5.
To generate equivalent expressions with positive exponents, we can use the properties of integer exponents. The properties we will use are:
1. A negative exponent represents the reciprocal of the base raised to the positive exponent.
2. When multiplying two numbers with the same base but different exponents, you can add the exponents.
3. When dividing two numbers with the same base but different exponents, you can subtract the exponents.
Now let's apply these properties to the given expression:
40 ⋅ 5^(-2)(-2)^(-3)
Using property 1, we can rewrite 5^(-2) as 1/5^2 and (-2)^(-3) as 1/(-2)^3.
After applying the property, the expression becomes:
40 ⋅ (1/5^2) ⋅ (1/(-2)^3)
Now let's simplify further by evaluating the exponents:
40 ⋅ (1/25) ⋅ (1/(-8))
To evaluate (-8), we can use property 1 again:
40 ⋅ (1/25) ⋅ (1/(-2^3))
Simplifying further, we have:
40 ⋅ (1/25) ⋅ (1/(-8))
Now, let's multiply the numerators and denominators:
(40 ⋅ 1 ⋅ 1) / (25 ⋅ -8)
Simplifying further, we get:
40 / (-200)
Finally, canceling out common factors, we simplify the expression to:
1 / (-5)
So, the equivalent expression is 1 / (-5).