Apply the Properties of Integer Exponents to generate equivalent expressions to 4 ^ 0 * 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, we can rewrite the expression using the following rules:
1. Any number raised to the power of zero is equal to 1: a^0 = 1
2. For any number a, a^(-n) = 1 / a^n (reciprocal)
Applying these rules to the expression 4^0 * 5^-2(-2)^-3, we get:
4^0 * 5^-2(-2)^-3 = 1 * 1 / (5^2 * 1 / (-2)^3)
= 1 / (5^2 * 1 / (-8))
= 1 / (25 * 1 / (-8))
= 1 / (25 / (-8))
= -8/25
Therefore, the answer is -8/25.
To apply the properties of integer exponents, let's break down the given expression step by step and simplify it using the following rules:
1. Rule: Any number raised to the power of 0 is equal to 1.
2. Rule: For any number a, a^(-n) = 1/a^n.
3. Rule: When multiplying exponential expressions with the same base, we add the exponents.
Given expression: 4^0 * 5^(-2(-2)^(-3))
Step 1: Simplify the exponent (-2)^(-3):
To simplify (-2)^(-3), we apply rule 2:
(-2)^(-3) = 1/(-2)^3
= 1/(-2 * -2 * -2)
= 1/(-8)
= -1/8
Step 2: Substitute the simplified exponent back into the expression:
4^0 * 5^(-2(-2)^(-3)) = 4^0 * 5^(-2 * -1/8)
Step 3: Simplify the exponents:
In our given expression, 4^0 is equal to 1 (using rule 1).
So, we're left with:
1 * 5^(-2 * -1/8) = 5^(2 * 1/8)
Step 4: Multiply the exponents:
5^(2 * 1/8) = 5^(2/8)
= 5^(1/4)
Step 5: Simplify the exponent:
In the final expression, 5^(1/4) means taking the fourth root of 5.
Therefore, the equivalent expression to 4^0 * 5^(-2(-2)^(-3)) with only positive exponents is 5^(1/4), which means the fourth root of 5.
To solve for the simplified fraction with no remaining exponents, we don't need to simplify further. Therefore, the answer is 5^(1/4), which represents the fourth root of 5.
To simplify the expression 4^0 * 5^-2(-2)^-3 and convert it into an equivalent expression with only positive exponents, we can use the following properties of integer exponents:
1. Any number raised to the power of zero is equal to 1: x^0 = 1, where x is any non-zero number.
2. The reciprocal of x^a is equal to x^-a: 1/(x^a) = x^-a.
3. When a negative exponent is applied to a number, we can take the reciprocal and make the exponent positive.
Now let's apply these properties step-by-step.
Step 1: Apply the first property (x^0 = 1) to simplify 4^0:
4^0 = 1
Step 2: Apply the third property to simplify 5^-2:
5^-2 = 1/(5^2) = 1/25
Step 3: Apply the third property to simplify (-2)^-3:
(-2)^-3 = 1/((-2)^3) = 1/(-2^3) = 1/(-8) = -1/8
Now that we've simplified the expression, we can rewrite it as follows:
1 * (1/25) * (-1/8)
Multiplying these fractions gives us:
(-1/25) * (1/8) = -1/200
Therefore, the simplified fraction for the expression 4^0 * 5^-2(-2)^-3 is -1/200.