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 simplified fraction.
To generate an equivalent expression with only positive exponents, we can apply the following properties of integer exponents:
1. Any non-zero number raised to the power of 0 is equal to 1: a^0 = 1 (where a is a non-zero number)
2. The reciprocal of a number raised to a negative exponent is equal to the number raised to the positive exponent in the denominator: a^(-n) = 1/(a^n)
Let's apply these properties step by step to simplify the expression:
First, using property 1, we have:
4^0 = 1
Now, let's apply property 2 to simplify the expression further:
5^(-2) = 1/(5^2) = 1/25
(-2)^(-3) = 1/((-2)^3) = 1/(-8) = -1/8
Now, substituting these simplified values back into the expression, we have:
1x(1/25)/( -1/8 )
Simplifying further, we have:
(1/25) × (-8/-1) = 8/25
Therefore, the simplified fraction equivalent to the expression 4^0x5^(-2)/(-2)^(-3) is 8/25.
To generate an equivalent expression with only positive exponents, we can rewrite the expression using the following properties of integer exponents:
1. To change a negative exponent to a positive exponent, we can take the reciprocal of the base and change the exponent sign to positive.
For example, a^(-n) = 1/a^n, where a ≠ 0.
Let's apply these properties step-by-step:
1. Applying the property, we can rewrite the expression with positive exponents:
4^0 * 5^(-2) / (-2)^(-3)
= 4^0 * 5^(-2) * (-2)^3
Note: Any base raised to the power of zero is equal to 1.
2. Simplifying further:
4^0 is equal to 1, so we can remove it from the expression:
1 * 5^(-2) * (-2)^3
= 5^(-2) * (-2)^3
3. We can expand the exponent of (-2)^3:
(-2)^3 = (-2) * (-2) * (-2) = -2 * -2 * -2 = -8
4. Now, plugging in the value of (-2)^3 back into the expression:
5^(-2) * (-8)
5. Applying the property of negative exponents:
5^(-2) = 1/5^2 = 1/25
6. Simplifying the expression further:
1/25 * (-8) = (-8)/25
Therefore, the equivalent expression 4^0 * 5^(-2) / (-2)^(-3) with only positive exponents simplifies to -8/25 as a simplified fraction.
To generate an equivalent expression with only positive exponents, we can apply the properties of integer exponents:
1. Property: Any non-zero number raised to the power of zero is equal to 1.
- Therefore, 4^0 is equal to 1.
2. Property: When you have a negative exponent, you can move the base to the denominator and change the exponent to positive.
- Therefore, 5^(-2) can be rewritten as 1/5^2.
3. Property: When you have a negative exponent in the denominator, you can move the base to the numerator and change the exponent to positive.
- Therefore, (-2)^(-3) can be rewritten as 1/(-2^3).
Now let's apply these properties to the given expression:
4^0 * 5^(-2) / (-2)^(-3)
Substituting with the properties:
1 * (1 / 5^2) / (1 / (-2^3))
Simplifying further:
1 * 1 / 5^2 / 1 / (-2^3)
Next, we perform the division:
1 * 1 * (-2^3) / (5^2 * 1)
Simplifying again:
(-2^3) / (5^2)
Now we calculate the powers:
(-8) / 25
Therefore, the equivalent expression to 4^0 * 5^(-2) / (-2)^(-3) with only positive exponents is (-8) / 25.
As a simplified fraction, the value of the expression is -8/25.