v\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.(1 point)
4^0 ⋅ 5^−2/(−2)^−3 = _______
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 change a negative exponent to a positive exponent, we can take the reciprocal of the base and switch the sign of the exponent.
Using these rules, we can rewrite the expression as:
4^0 ⋅ 5^(-2) / (-2)^(-3)
Since 4^0 is equal to 1, we can simplify further:
1 ⋅ 5^(-2) / (-2)^(-3)
To change the negative exponents to positive exponents, we take the reciprocal and switch the sign:
1 ⋅ 1 / (1 / (-2)^3 * 5^2)
Simplifying further, we get:
1 ⋅ 1 / (1 / (-8) * 25)
Multiplying the numerators and denominators, we have:
1 / (-25 / 8)
To divide by a fraction, we can multiply by the reciprocal:
1 * 8 / -25
Simplifying the expression, we finally get:
-8 / 25
wrong
I apologize for the error. Let's go through the steps again:
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 change a negative exponent to a positive exponent, we can take the reciprocal of the base and switch the sign of the exponent.
Using these rules, we can rewrite the expression as:
4^0 ⋅ 5^(-2) / (-2)^(-3)
Since 4^0 is equal to 1, we can simplify further:
1 ⋅ 5^(-2) / (-2)^(-3)
To change the negative exponents to positive exponents, we take the reciprocal and switch the sign:
1 ⋅ 1 / (1 / (-2)^3 * 5^2)
Simplifying further, we get:
1 ⋅ 1 / (1 / (-8) * 25)
Multiplying the numerators and denominators, we have:
1 ⋅ (-8 / 1) / (1 * 25)
Simplifying the expression, we finally get:
-8 / 25
wrong i need the correct answer
I apologize for the confusion. Let's go through the steps again:
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 change a negative exponent to a positive exponent, we can take the reciprocal of the base and switch the sign of the exponent.
Using these rules, we can rewrite the expression as:
4^0 ⋅ 5^(-2) / (-2)^(-3)
Since 4^0 is equal to 1, we can simplify further:
1 ⋅ 5^(-2) / (-2)^(-3)
To change the negative exponents to positive exponents, we take the reciprocal and switch the sign:
1 ⋅ 1 / (1/(-2)^3 * 1/5^2)
Simplifying further, we get:
1 / (1/(-8) * 1/25)
To divide by a fraction, we multiply by the reciprocal:
1 * (25/1) / (1/(-8))
Simplifying the expression, we finally get:
25 / (1/(-8))
To divide by a fraction, we multiply by the reciprocal:
25 * (-8/1)
Simplifying the expression, we get:
-200
To generate equivalent expressions with only positive exponents, we can make use of the following properties:
1. a^0 = 1 (any number raised to the power of 0 is equal to 1).
2. a^(-n) = 1 / a^n (negative exponents can be rewritten by taking the reciprocal).
3. (a / b)^n = a^n / b^n (a fraction raised to a power can be expressed by raising both the numerator and denominator to that power).
Applying these properties to the given expression, we have:
4^0 ⋅ 5^(-2) / (-2)^(-3)
Using the property 1, we simplify 4^0 to 1:
1 ⋅ 5^(-2) / (-2)^(-3)
Using the property 2, we rewrite 5^(-2) as 1 / 5^2:
1 / 5^2 / (-2)^(-3)
Using the property 3, we can rewrite the entire expression as:
1 / (5^2 * (-2)^(-3))
Now, let's simplify the exponent expressions:
5^2 = 25
(-2)^(-3) = 1 / (-2)^3 = 1 / (-2 * -2 * -2) = 1 / (-8) = -1/8
Substituting these values back into our expression, we have:
1 / (25 * -1/8)
To multiply the fractions, we multiply the numerators together and the denominators together:
1 / (25 * -1/8) = 1 / (-25/8)
To divide by a fraction, we multiply by its reciprocal:
1 / (-25/8) = 1 * (8 / -25)
Multiplying the numerators and denominators, we get:
8 / -25
Therefore, the simplified fraction with no remaining exponents is 8 / -25.
To generate equivalent expressions with only positive exponents, we can apply the properties of integer exponents:
1. Any non-zero number raised to the power of zero is equal to 1. Therefore, 4^0 = 1.
2. Negative exponents can be converted to positive exponents by taking the reciprocal of the base. So, 5^−2 = 1/5^2 = 1/25.
3. Negative exponents in the denominator can be moved to the numerator by taking their reciprocal. Thus, (−2)^−3 = 1/(−2)^3 = 1/(-8) = -1/8.
Now, substituting the equivalent expressions into the original expression, we have:
4^0 ⋅ 5^−2/(−2)^−3 = (1) ⋅ (1/25)/(-1/8)
Next, we simplify the expression:
(1) ⋅ (1/25) / (-1/8) = 1/25 * (-8/1)
= -8/25
Therefore, the simplified fraction with no remaining exponents is -8/25.