Apply the properties of integer exponents to generate equivalent expressions to 4^0x5^-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 use the following rules:
Rule 1: Any number raised to the power of 0 is equal to 1.
Rule 2: When dividing two numbers with the same base, subtract the exponents.
Rule 3: When multiplying two numbers with the same base, add the exponents.
Rule 4: When raising a power to another power, multiply the exponents.
Using these rules, let's apply them step by step to simplify the expression:
Step 1: Apply the first rule to 4^0 and 5^-2:
4^0 * 5^-2 = 1 * (1/5^2)
Step 2: Combine the fractions:
1 * (1/5^2) = 1/5^2
Step 3: Apply the first rule again:
1/5^2 = 1/(5*5)
Step 4: Simplify the expression:
1/(5*5) = 1/25
So, the equivalent expression with only positive exponents is 1/25.
Now, solving the expression:
(-2)^3 = (-2) * (-2) * (-2) = -8
Finally, dividing 1/25 by -8:
(1/25) / (-8) = 1/25 * (-1/8) = -1/200
Therefore, the solution to the expression 4^0 * 5^-2 / (-2)^3 is -1/200.
To apply the properties of integer exponents and generate equivalent expressions with only positive exponents, we can start by simplifying the expression step by step:
1. Start with the expression: 4^0 * 5^(-2) * (-2)^3.
2. The property of any number (except zero) raised to the power of 0 is equal to 1. So 4^0 equals 1.
3. Substitute 4^0 back into the expression: 1 * 5^(-2) * (-2)^3.
4. The property of negative exponents states that a number raised to a negative exponent is equal to 1 over the number raised to the positive value of that exponent. Therefore, 5^(-2) is equivalent to 1 / 5^2.
5. Substitute 5^(-2) back into the expression: 1 * (1 / 5^2) * (-2)^3.
6. The exponent rule for negative numbers states that (-a)^n is equal to (-1)^n * a^n. Therefore, (-2)^3 becomes (-1)^3 * 2^3.
7. Simplify (-1)^3 to -1: 1 * (1 / 5^2) * (-1) * 2^3.
8. Simplify 2^3 to 8: 1 * (1 / 5^2) * (-1) * 8.
9. Simplify 5^2 to 25: 1 * (1 / 25) * (-1) * 8.
10. Multiply the constants together: -8 / 25.
So, the simplified fraction form of the expression 4^0 * 5^(-2) * (-2)^3 is -8/25.