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 =
Using the properties of integer exponents, we have:
4^0 ⋅ 5^−2 / (−2)^−3
Since any number raised to the power of 0 is equal to 1, we can simplify 4^0 to 1:
1 ⋅ 5^−2 / (−2)^−3
Since any number raised to a negative exponent is equal to its reciprocal raised to the positive exponent, we can rewrite 5^−2 as 1/5^2:
1 ⋅ 1/(5^2) / (−2)^−3
Similarly, we can rewrite (−2)^−3 as 1/(−2)^3:
1 ⋅ 1/(5^2) / 1/(−2)^3
Now, we can apply the division of fractions rule, which states that dividing by a fraction is the same as multiplying by its reciprocal. So we can rewrite the expression as:
1 ⋅ 1/(5^2) * (−2)^3
Simplifying further, we have:
1/(25) * (−8)
Multiplying the numerators and denominators, we get:
-8/25
Thus, the equivalent expression is -8/25.
wrong
Apologies for the confusion in my previous response. Let's go through the steps again to find the correct answer.
Using the properties of integer exponents, we have:
4^0 ⋅ 5^−2 / (−2)^−3
Since any number raised to the power of 0 is equal to 1, we can simplify 4^0 to 1:
1 ⋅ 5^−2 / (−2)^−3
Next, we can rewrite 5^−2 as 1/5^2 and (−2)^−3 as 1/(−2)^3:
1 ⋅ 1/(5^2) / 1/(−2)^3
Now, let's apply the property of negative exponents to move the reciprocal of the base with the negative exponent to the numerator:
1 ⋅ (−2)^3 / (5^2)
Simplifying further, we have:
−8 / 25
Thus, the equivalent expression is -8/25.
you just copied and pasted the same answer WRONG
Sorry for the repeated mistake. Let's go through the steps again to find the correct answer.
Using the properties of integer exponents, we have:
4^0 ⋅ 5^−2 / (−2)^−3
Since any number raised to the power of 0 is equal to 1, we can simplify 4^0 to 1:
1 ⋅ 5^−2 / (−2)^−3
Next, we can rewrite 5^−2 as 1/5^2 and (−2)^−3 as 1/(−2)^3:
1 ⋅ 1/(5^2) / 1/(−2)^3
Now, let's apply the properties of reciprocals and negative exponents:
1 / (5^2) / (−2)^3
Since (-2)^3 is equal to -8, we can replace it in the expression:
1 / (5^2) / (-8)
We can simplify further by squaring 5:
1 / 25 / (-8)
To divide by a fraction, we can multiply by its reciprocal:
1 * (-8) / 25
Simplifying the numerator, we have:
-8 / 25
Thus, the equivalent expression is -8/25.
To generate an equivalent expression with only positive exponents, we can use the properties of integer exponents:
1. Any number raised to the power of 0 is equal to 1, so we can replace 4^0 with 1.
Now the expression becomes:
1 ⋅ 5^−2 / (−2)^−3
2. To remove the negative exponents, we can use the property: a^−n = 1 / a^n. So we can rewrite 5^−2 as 1 / 5^2 and (−2)^−3 as 1 / (−2)^3.
The expression becomes:
1 ⋅ 1 / 5^2 / 1 / (−2)^3
3. Division of fractions can be done by multiplying the numerator by the reciprocal of the denominator. So, we can rewrite the expression as:
1 ⋅ (−2)^3 / 5^2
4. Now we can simplify the expression by evaluating the exponents:
1 ⋅ (−2)^3 = 1 ⋅ (-8) = -8
5^2 = 5 * 5 = 25
The expression simplifies to:
-8 / 25
So, the final answer is -8/25.
To apply the properties of integer exponents and generate equivalent expressions with only positive exponents, we can use the following rules:
1. Any number raised to the power of 0 is equal to 1.
2. For any nonzero number a, a^(-n) is equal to 1/a^n.
3. When dividing two numbers with the same base but different exponents, we subtract the exponents.
Now let's apply these rules to simplify the expression:
4^0 ⋅ 5^(-2) / (-2)^(-3)
Since 4^0 is equal to 1, the expression becomes:
1 ⋅ 5^(-2) / (-2)^(-3)
Applying the rule from #2, we rewrite 5^(-2) as 1/5^2:
1 ⋅ (1/5^2) / (-2)^(-3)
Now, applying the rule from #2 again, we rewrite (-2)^(-3) as 1/(-2)^3:
1 ⋅ (1/5^2) / (1/(-2)^3)
For simplicity, we can multiply the numerator and the denominator by (-2)^3:
(1 ⋅ (1/5^2) ⋅ (-2)^3) / (1)
Simplifying the numerator:
1 ⋅ (1/25) ⋅ (-8)
Multiplying the fractions:
-8/25
Therefore, the expression 4^0 ⋅ 5^(-2) / (-2)^(-3) simplifies to -8/25, which is a simplified fraction with no remaining exponents.