Which of the following is an equivalent expression to 15^0 ⋅ 7^−2/(−4)^−3 with only positive exponents that has been generated by applying the Properties of Integer Exponents?
Responses
4^3/7^2
1/7^2 ⋅ (−4)^−3
7^2/(−4)^3
(−4)^3/7^2
1/7^2 ⋅ (−4)^−3
wrong
I apologize for the incorrect response. The correct equivalent expression with only positive exponents is:
1 / (15^0 * 7^2 * (-4)^3)
To find an equivalent expression with only positive exponents, we can apply the properties of integer exponents. Let's break down the given expression step by step:
Expression: 15^0 ⋅ 7^(-2)/(−4)^(-3)
Step 1: Any non-zero number raised to the power of 0 is equal to 1. Thus, 15^0 equals 1.
Expression: 1 ⋅ 7^(-2)/(−4)^(-3)
Step 2: To convert a negative exponent to a positive exponent, we can move the base to the opposite side of the fraction and change the sign of the exponent. So, 7^(-2) becomes 1/7^2 and (−4)^(-3) becomes 1/(-4)^3.
Expression: 1 ⋅ 1/7^2 / 1/(-4)^3
Step 3: When dividing fractions, we multiply the first fraction by the reciprocal of the second fraction. So, 1/7^2 divided by 1/(-4)^3 is the same as 1/7^2 * (-4)^3/1.
Expression: 1 * (-4)^3 / 7^2
Step 4: Simplify the expression. (-4)^3 equals -64.
Expression: -64 / 7^2
Therefore, an equivalent expression with only positive exponents is: -64 / 7^2
To simplify the expression and generate an equivalent expression with only positive exponents, we can use the properties of integer exponents.
First, recall that any number raised to the power of zero is equal to 1. So, we can simplify 15^0 to 1.
Next, when we have a negative exponent, we can move the base to the denominator and change the sign of the exponent. This means that 7^(-2) can be written as 1/7^2.
Similarly, (-4)^(-3) can be expressed as 1/(-4)^3.
Now we can rewrite the original expression as:
1 ⋅ (1/7^2) / (1/(-4)^3)
Multiplying two fractions is the same as multiplying the numerators and multiplying the denominators. So, we have:
1/1 ⋅ (1/7^2) / (1/(-4)^3)
Simplifying the expression further, we can remove the 1s in the numerator and denominator:
(1/7^2) / (1/(-4)^3)
To divide by a fraction, we can multiply by its reciprocal. Therefore, we can rewrite the expression as:
(1/7^2) ⋅ ((-4)^3/1)
Multiplying the numerators and the denominators, we get:
((-4)^3) / (7^2 ⋅ 1)
Finally, simplifying further, we have:
(−4)^3 / 7^2
So, the equivalent expression with only positive exponents, generated using the properties of integer exponents, is (−4)^3 / 7^2.