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?(1 point)
Responses
7^2\(−4)^3
(−4)^3\7^2
1\7^2 ⋅ (−4)^−3
4^3\7^2
To simplify the expression, we can use the properties of exponents.
First, we know that any number raised to the power of 0 is equal to 1. So, 15^0 is equal to 1.
Next, we can rewrite 7^(-2) as 1/7^2. The negative exponent indicates that we need to take the reciprocal of the base.
Similarly, (-4)^(-3) can be rewritten as 1/(-4)^3.
Combining these simplifications, we get:
15^0 * 7^(-2) * (-4)^(-3)
= 1 * 1/7^2 * 1/(-4)^3
= 1/7^2 * 1/(-64)
= 1/49 * 1/(-64)
= 1/(49 * -64)
So, the equivalent expression with positive exponents is 1/(49 * -64), which is not one of the given options.
To find an equivalent expression with positive exponents, we can use the properties of exponents.
Starting with 15^0 ⋅ 7^−2 ⋅ (−4)^−3, we know that any number raised to the power of 0 is equal to 1. So, 15^0 becomes 1.
Now, we can rewrite the expression as:
1 ⋅ 7^−2 ⋅ (−4)^−3
Using the property of negative exponents, any number raised to a negative exponent can be moved to the denominator by changing the sign of the exponent.
So, 7^−2 becomes 1/7^2 and (−4)^−3 becomes 1/(−4)^3.
The expression now becomes:
1/(7^2) ⋅ 1/(−4)^3
Simplifying further, (−4)^3 is equal to 4^3:
1/(7^2) ⋅ 1/4^3
Therefore, the equivalent expression with positive exponents is:
1/7^2 ⋅ 4^3
So, the correct option is 1/7^2 ⋅ 4^3.