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)
First, let's apply the properties of integer exponents to 40 ⋅ 5^(-2)(-2)^(-3):
1. Negative exponent property: a^(-n) = 1/a^n
40 ⋅ 5^(-2)(-2)^(-3) = 40/(5^2),(2^3)
= 40/(5^2)(2^3)
2. Product of powers property: a^n ⋅ a^m = a^(n+m)
40/(5^2)(2^3) = 40/((5^2)⋅(2^3))
Now, let's simplify the expression:
1. Calculate the exponent of each base:
(5^2) = 25
(2^3) = 8
2. Simplify the expression:
40/((5^2)⋅(2^3)) = 40/(25⋅8)
3. Calculate the value of 25⋅8:
25⋅8 = 200
4. Simplify the expression further:
40/200 = 1/5
Therefore, the simplified fraction with no remaining exponents is 1/5.
my apologies for the confusion
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)
To apply the properties of integer exponents and generate equivalent expressions, we can rewrite the expression as follows:
40 ⋅ 5^(-2)(-2)^(-3)
Using the properties of exponents, we have:
1. Negative exponent property: a^(-n) = 1/a^n
Applying this property to 5^(-2), we get:
5^(-2) = 1/5^2
2. Negative exponent property: a^(-n) = 1/a^n
Applying this property to (-2)^(-3), we get:
(-2)^(-3) = 1/(-2)^3
Now, let's rewrite the expression using these equivalent forms:
40 ⋅ 5^(-2)(-2)^(-3) = 40 ⋅ (1/5^2) ⋅ (1/(-2)^3)
Next, we simplify the exponents:
1/5^2 = 1/25
1/(-2)^3 = 1/(-2 * -2 * -2) = 1/(-8) = -1/8
Now, substitute these simplified forms back into the expression:
40 ⋅ (1/5^2) ⋅ (1/(-2)^3) = 40 ⋅ (1/25) ⋅ (-1/8)
Next, multiply the numerators and denominators:
40 ⋅ (1/25) ⋅ (-1/8) = (40 ⋅ 1 ⋅ -1) / (25 ⋅ 8) = -40 / 200 = -1/5
Thus, the simplified fraction with no remaining exponents is -1/5.
To generate equivalent expressions with only positive exponents for the expression 40 ⋅ 5^(-2)(-2)^(-3), we can use the following properties of integer exponents:
1. Product of powers:
a^m * a^n = a^(m+n)
2. Quotient of powers:
a^m / a^n = a^(m-n)
3. Power of a power:
(a^m)^n = a^(m*n)
4. Negative exponent property:
a^(-n) = 1/a^n
Now applying these properties step by step:
Step 1: Simplify the negative exponents:
40 ⋅ 5^(-2)(-2)^(-3)
= 40 * 1/5^2 * 1/(-2)^3
= 40 * 1/25 * 1/(-2)^3
Step 2: Simplify the expression inside the parentheses:
= 40 * 1/25 * 1/(-8)
= 40/25 * 1/(-8)
Step 3: Simplify the fractions:
= (40/25) * (1/(-8))
= (8/5) * (1/(-8))
Step 4: Simplify the expression:
= (8/5) * (-1/8)
= -1/5
Therefore, the simplified fraction of the expression 40 ⋅ 5^(-2)(-2)^(-3) is -1/5.
To apply the properties of integer exponents, we can use the following rules:
1. Product Rule: \(a^m \cdot a^n = a^{m+n}\)
2. Quotient Rule: \(\frac{{a^m}}{{a^n}} = a^{m-n}\)
3. Power Rule: \((a^m)^n = a^{mn}\)
4. Negative Exponent Rule: \(a^{-n} = \frac{1}{{a^n}}\)
Now let's apply these rules to the given expression step by step:
Step 1: Apply the negative exponent rule
\(40 \cdot 5^{-2(-2)-3} = 40 \cdot 5^{-2 \cdot (-2)-3} = 40 \cdot 5^{4-3}\)
Step 2: Simplify the exponent
\(5^{4-3} = 5^1\)
Step 3: Evaluate the expression
\(40 \cdot 5^1 = 40 \cdot 5 = 200\)
So, the equivalent expression to \(40 \cdot 5^{-2(-2)-3}\) with only positive exponents is \(200\).