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 use the following rules:
1. Any number raised to the power of 0 is equal to 1.
2. To divide two numbers with like bases but different exponents, subtract the exponents and keep the base the same.
3. To raise a power to another power, multiply the exponents.
Now, let's apply these properties to simplify the expression:
4^0 * 5^-2 * (-2)^3
Since 4^0 equals 1, we can rewrite the expression as:
1 * 5^-2 * (-2)^3
Next, we can use the third property to simplify (-2)^3:
1 * 5^-2 * (-2)^3 = 1 * 5^-2 * (-2 * -2 * -2)
= 1 * 5^-2 * (-8)
Now, let's use the second property to rewrite 5^-2:
= 1 * (1/5^2) * (-8)
= 1 * (1/25) * (-8)
Finally, we can multiply the fractions:
= -8/25
Therefore, the simplified fraction for the expression 4^0 * 5^-2 * (-2)^3 with only positive exponents is -8/25.
To generate equivalent expressions with only positive exponents, we can apply the following properties of integer exponents:
1. The power of a product property: (a * b) ^ n = a ^ n * b ^ n
2. The power of a power property: (a ^ m) ^ n = a ^ (m * n)
3. The power of a reciprocal property: (a / b) ^ n = a ^ n / b ^ n
4. Any number raised to the power of 0 is equal to 1: a ^ 0 = 1
Now let's apply these properties step by step:
1. 4^0 * 5^-2 \ (-2)^3
2. Applying the power of a number raised to 0 is equal to 1 property:
1 * 5^-2 \ (-2)^3
3. Apply the power of a reciprocal property to the expression 5^-2:
1 * (1/5^2) \ (-2)^3
4. Simplify the reciprocal expression 1/ 5^2:
1 * 1/25 \ (-2)^3
5. Apply the power of a number raised to an exponent property:
1 * 1/25 \ -2^3
6. Evaluate the exponent -2^3 first, since exponentiation has higher precedence than any other operation:
1 * 1/25 \ (-8)
7. Simplify the expression 1 * 1/25:
1/25 \ (-8)
8. Divide 1 by 25:
1/25 \ (-8)
9. To divide fractions, we can multiply by the reciprocal. So, multiply 1/25 by -1/8:
(1/25) * (-1/8)
10. Multiply the numerators together: 1 * -1 = -1
11. Multiply the denominators together: 25 * 8 = 200
12. Simplify the fraction -1/200
Therefore, the expression 4^0 * 5^-2 \ (-2)^3 simplifies to -1/200.