Apply the Properties of integer exponents to generate an equivalent expression to 4^0x5^(-2)/(-2)^(-3) with only positive exponents, then solve the expression. As a whole number.
To generate an equivalent expression with only positive exponents, we can use the following properties of integer exponents:
1. Property 1: Any non-zero number raised to the power of 0 is equal to 1.
2. Property 2: A negative exponent can be converted to a positive exponent by taking its reciprocal.
Applying these properties, we can rewrite the expression as follows:
4^0 x 5^(-2) / (-2)^(-3) = 1 x (1/5^2) / (1/(-2)^3)
First, let's simplify the expression inside the parentheses:
= 1 x (1/25) / (1/(-8))
Now, let's simplify the division:
= 1 x (1/25) x (-8/1)
= -(8/25)
So, the equivalent expression is -(8/25).
Since we are looking for the solution as a whole number, we can see that the solution is -8 divided by 25, which gives us -0.32. However, -0.32 is not a whole number.
Therefore, there is no solution to the given expression as a whole number.
To apply the properties of integer exponents and generate an equivalent expression with only positive exponents, we can use the following rules:
1. a^0 = 1 (any number raised to the power of 0 is equal to 1).
2. a^(-n) = 1/a^n (a negative exponent is equivalent to the reciprocal of the positive exponent).
Using these rules, let's break down the given expression step by step:
4^0 x 5^(-2) / (-2)^(-3)
Step 1: Apply the property of a^0:
4^0 x 5^(-2) / (-2)^(-3) = 1 x 5^(-2) / (-2)^(-3)
Step 2: Apply the property of a negative exponent:
1 x 5^(-2) / (-2)^(-3) = 1 x 1/5^2 / 1/(-2)^3
Step 3: Simplify the exponents:
1 x 1/5^2 / 1/(-2)^3 = 1 x 1/25 / 1/(-8)
Step 4: Simplify the divisions:
1 x 1/25 / 1/(-8) = 1 x (-8) / 25
Step 5: Compute the result by multiplying:
1 x (-8) / 25 = -8 / 25
Therefore, the simplified expression is -8/25.
To generate an equivalent expression with only positive exponents, we need to apply the properties of integer exponents.
We know that any nonzero number raised to the power of 0 is equal to 1 (a^0 = 1).
Let's start with the given expression:
4^0 * 5^(-2) / (-2)^(-3)
Applying the property, 4^0 is equivalent to 1:
1 * 5^(-2) / (-2)^(-3)
Next, we know that negative exponents can be rewritten as positive exponents by taking the reciprocal of the base and changing the sign of the exponent (a^(-b) = 1 / a^b).
So, 5^(-2) can be written as 1 / 5^2, and (-2)^(-3) can be written as 1 / (-2)^3.
Substituting these values into the expression, we get:
1 * (1 / 5^2) / (1 / (-2)^3)
Simplifying further, we can rewrite the expression as:
1 * (1 / 25) / (1 / (-8))
Division is equivalent to multiplying by the reciprocal of the divisor (a / b = a * (1 / b)). So, we can rewrite the expression as:
1 * (1 / 25) * (-8 / 1)
Now, we can multiply the numerators and denominators respectively:
1 * -8 / (25 * 1)
Simplifying the expression, we have:
-8 / 25
Therefore, the equivalent expression with only positive exponents is -8/25, and the solution is -8/25 as a whole number.