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
7^2(−4)^3
4^3/7^2
(−4)^3/7^2
1/7^2 ⋅ (−4)^−3
To apply the properties of integer exponents, we will use the following rules:
1. Any base raised to the power of 0 is equal to 1.
2. When dividing two exponents with the same base, subtract the exponents.
3. When raising a negative number to an even exponent, the result is positive.
4. When raising a negative number to an odd exponent, the result is negative.
Using these rules, we can simplify the expression:
15^0 ⋅ 7^−2/(−4)^−3
Applying rule 1 to 15^0 (any number raised to the power of 0 is 1), we get:
1 ⋅ 7^−2/(−4)^−3
Applying rule 3 to 7^−2 (raising a positive number to a negative exponent), we get:
1/(7^2 ⋅ (−4)^−3)
Applying rule 4 to (−4)^−3 (raising a negative number to an odd exponent), we get:
1/(7^2 ⋅ (−1)^3 ⋅ 4^−3)
Simplifying (−1)^3 as -1 and 4^−3 as 1/4^3, we get:
1/(7^2 ⋅ (−1) ⋅ 1/4^3)
Simplifying further:
1/((49)(-1)(1/64))
Calculating the values:
1/(-3136/64)
Simplifying:
64/(-3136)
Applying rule 4 to the negative number, we get a positive result:
64/3136
Therefore, the equivalent expression with only positive exponents is:
4^3/7^2. Answer: \boxed{4^3/7^2}.
To simplify the expression with positive exponents, we need to apply the properties of integer exponents.
First, let's simplify each term separately.
15^0 is equal to 1, since any number (except 0) raised to the power of 0 is equal to 1.
7^(-2) is equal to 1/7^2, since a negative exponent indicates the reciprocal of the base raised to the positive exponent.
(-4)^(-3) is equal to 1/(-4)^3, again because a negative exponent indicates the reciprocal.
Now, let's simplify the expression:
15^0 ⋅ 7^(-2) / (-4)^(-3)
This becomes:
1 ⋅ 1/7^2 / 1/(-4)^3
Simplifying further:
1 ⋅ 1/7^2 ⋅ (-4)^3
Since 1 multiplied by any number is equal to that number, and (-4)^3 is equal to -64, we get:
1/7^2 ⋅ (-64)
Finally, since multiplying a fraction by -1 only changes its sign, we have:
-64/7^2
So, the equivalent expression with only positive exponents is the last option:
1/7^2 ⋅ (-4)^(-3) is equivalent to -64/7^2.
To simplify the expression and convert all exponents to positive, we can apply the following properties of integer exponents:
1. Any number raised to the power of 0 is equal to 1.
2. To change a negative exponent to a positive exponent, we can take the reciprocal of the base raised to the positive exponent.
Let's break down the given expression step by step:
15^0 ⋅ 7^−2/(−4)^−3
Step 1: Start with the base raised to the power of 0, which equals 1:
15^0 = 1
Step 2: Convert the negative exponent by taking the reciprocal of the base raised to the positive exponent:
7^−2 = 1/7^2
(−4)^−3 = 1/(−4)^3
Step 3: Apply the properties of integer exponents to the simplified expression:
1 ⋅ 1/7^2/(−4)^3
Now, we need to rearrange the terms to make the exponents positive:
1/(7^2 ⋅ (−4)^3)
Therefore, the equivalent expression to 15^0 ⋅ 7^−2/(−4)^−3 with only positive exponents is 1/(7^2 ⋅ (−4)^3), which is option D: 1/7^2 ⋅ (−4)^−3.
wrong
I apologize for the incorrect response. Let's attempt to simplify the expression correctly.
Given expression: 15^0 ⋅ 7^−2/(−4)^−3
First, we know that any number raised to the power of 0 is equal to 1.
So, 15^0 = 1.
Next, to convert the negative exponents to positive exponents, we take the reciprocal of the base raised to the positive exponent.
Thus, 7^−2 = 1/7^2 and (−4)^−3 = 1/(−4)^3 or 1/4^3.
Substituting these values back into the expression, we have:
1 ⋅ 1/7^2 / 1/4^3
To divide 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 multiplied by 4^3/1.
Multiplying the fractions gives us:
1/(7^2) * (4^3/1)
Simplifying further:
1/49 * 64/1
Multiplying these fractions gives us:
64/49
Therefore, the simplified expression with only positive exponents is 64/49.