15^0*7^-2/(-4)^-3
To solve this expression, we follow the order of operations, which is parentheses, exponents, multiplication/division (from left to right), and addition/subtraction (from left to right).
First, we evaluate any parentheses, but there are none in this expression.
Next, we evaluate any exponents.
15^0 = 1, since any number raised to the power of 0 equals 1.
Now the expression becomes: 1 * 7^-2 / (-4)^-3.
Next, we perform multiplication/division from left to right.
7^-2 = 1/7^2 = 1/49, since negative exponents indicate taking the reciprocal.
So, the expression becomes: 1 * 1/49 / (-4)^-3.
Now, let's evaluate the denominator, (-4)^-3.
(-4)^-3 = 1/(-4)^3 = 1/(-4 * -4 * -4) = 1/(-64) = -1/64.
So, the expression becomes: 1 * 1/49 / (-1/64).
When dividing by a fraction, we can multiply by its reciprocal. Therefore, dividing by -1/64 is the same as multiplying by -64/1.
Our expression becomes: 1 * 1/49 * (-64/1).
Now, we can multiply the numbers.
1 * 1 = 1, and 49 * 64 = 3136.
Therefore, the expression simplifies to: 1/3136.
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?
To rewrite the expression with positive exponents, we can apply the properties of integer exponents.
First, let's rewrite the expression: 15^0 ⋅ 7^−2/ (−4)^−3.
Since any number raised to the power of 0 equals 1, we have: 1 ⋅ 7^−2/ (−4)^−3.
Now, let's convert the negative exponents to positive exponents.
The reciprocal rule states that a^−n = 1/a^n. Therefore, we can rewrite 7^−2 as 1/7^2, and (−4)^−3 as 1/(-4)^3.
Thus, our expression becomes: 1 ⋅ 1/7^2 / 1/(-4)^3.
Next, we can simplify further by dividing by a fraction, which is the same as multiplying by the reciprocal of the denominator. In this case, we multiply by 1/1/(-4)^3.
So, our expression simplifies to: 1 ⋅ 1/7^2 ⋅ (-4)^3.
To simplify further, we can apply the power of a product rule. This rule states that (ab)^n = a^n * b^n. Therefore, we can rewrite the expression as:
1 ⋅ (1^3 / 7^2) ⋅ (-4)^3.
Simplifying further, we have: 1 ⋅ 1 / 49 ⋅ (-64).
Finally, multiplying the numbers together, we get: 1 / 3136.
Therefore, the expression 15^0 ⋅ 7^−2/ (−4)^−3 with only positive exponents is 1 / 3136.