which of the following is an equivalent expression to 15^0 x 7^-2/(-4)^-3 with only positive exponents that have been generated by applying the Properties of Integer Exponents?
a. 4^3/7^2
b. 1/7^2 x (-4(^-3
c. 7^2/(-4)^3
d. (-4)^3/7^2
To simplify the expression 15^0 x 7^-2/(-4)^-3 and convert all exponents to positive, you can use the Properties of Integer Exponents. Recall that any number raised to the power of 0 is equal to 1.
First, apply the property that any number raised to the power of 0 is 1. So, 15^0 becomes 1.
Next, apply the property of negative exponents. Any number raised to a negative exponent can be rewritten by taking the reciprocal of the base and changing the sign of the exponent. Therefore, 7^-2 becomes 1/7^2, and (-4)^-3 becomes 1/(-4)^3.
Now, the expression becomes: 1 x 1/7^2 / 1/(-4)^3.
To simplify further, multiply by the reciprocal of the denominator: 1 x 1/7^2 x (-4)^3.
So, the equivalent expression with only positive exponents generated by applying the Properties of Integer Exponents is:
d. (-4)^3/7^2.
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To simplify the expression 15^0 x 7^-2/(-4)^-3 with only positive exponents using the properties of integer exponents, we need to apply the following rules:
1. Any number raised to the power of 0 is equal to 1.
2. A negative exponent can be written as the reciprocal of the positive exponent.
Let's simplify each part step-by-step:
1. 15^0 = 1
2. 7^-2 = 1/7^2 (by applying rule 2)
3. (-4)^-3 = 1/(-4)^3 (by applying rule 2)
Now, the expression becomes:
1 x 1/7^2 x 1/(-4)^3
Multiplying the numerators and denominators together, we get:
1/7^2 x 1/(-4)^3 = 1/(7^2 * (-4)^3) = 1/(49 * (-64))
Simplifying further, we have:
1/(49 * (-64)) = 1/(-3136) = -1/3136
So, the equivalent expression with only positive exponents is:
d. (-4)^3/7^2
To find an equivalent expression to 15^0 x 7^-2/(-4)^-3 with only positive exponents, we can use the properties of integer exponents. Let's break down the process step by step:
1. Start with the given expression: 15^0 x 7^-2/(-4)^-3.
2. Since any number raised to the power of 0 is equal to 1, we can simplify 15^0 to 1. So, the expression becomes: 1 x 7^-2/(-4)^-3.
3. To get rid of the negative exponents, we can apply the property that says if we have a negative exponent, we can move the base to the denominator and change the exponent to positive. Therefore, the expression becomes: 1 x 1/7^2 / (1/(-4)^3).
4. Simplify the expression further by evaluating the positive exponents: 1 x 1/49 / (1/(-4)^3).
5. Simplify the denominators by performing the necessary calculations: 1 x 1/49 / 1/(-64).
6. To divide by a fraction, we can multiply by its reciprocal. Therefore, the expression becomes: 1 x 1/49 x (-64)/1.
7. Evaluate the multiplication: 1 x (-64)/49.
8. Simplify the expression: -64/49.
Therefore, the equivalent expression with only positive exponents is option (d) (-4)^3/7^2.