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?
Answers
4^3/7^2
1/7^2 ⋅ (−4)^−3
7^2/(−4)^3
(−4)^3/7^2
The equivalent expression with only positive exponents can be found by applying the property of integers exponents that states that a negative exponent can be rewritten as the reciprocal of the base raised to the positive exponent.
Applying this property to the given expression, we can rewrite 7^−2 as 1/7^2 and (−4)^−3 as 1/(−4)^3.
Therefore, the equivalent expression with only positive exponents is 15^0 ⋅ 1/7^2 ⋅ 1/(−4)^3.
Simplifying this expression, we get (−4)^3/7^2.
So, the correct answer is (−4)^3/7^2.
To simplify the expression and apply the Properties of Integer Exponents, follow these steps:
Step 1: Start by simplifying the expression 15^0. Any number raised to the power of 0 is equal to 1. So, 15^0 = 1.
Step 2: Rewrite the expression with only positive exponents. The expression becomes:
1 ⋅ 7^(-2) / (-4)^(-3)
Step 3: Apply the property of negative exponents. When a number with a negative exponent is moved to the denominator, the exponent becomes positive. The expression becomes:
1 / (7^(2)) / ( (-4)^(3) )
Step 4: Simplify the exponents:
1 / 7^2 / (-4)^3
Step 5: Simplify 7^2 by multiplying 7 * 7 = 49:
1 / 49 / (-4)^3
Step 6: Simplify (-4)^3 by multiplying -4 * -4 * -4 = -64:
1 / 49 / -64
Step 7: To divide by a fraction, we invert the fraction and multiply:
1 / 49 * (-64)
Step 8: Multiply to get the final answer:
-64/49
So, the equivalent expression with only positive exponents is (-64/49).
Therefore, none of the given answers are equivalent expressions to 15^0 ⋅ 7^-2 / (-4)^-3 with only positive exponents.
To simplify the given expression and convert all exponents to positive, we can apply the following properties of integer exponents:
1. Any non-zero number raised to the power of zero is equal to 1.
2. For a positive base, when dividing powers with the same base, we subtract the exponents.
3. For a negative base, when dividing powers with the same base, we add the exponents.
Let's break down the given expression step by step:
15^0 ⋅ 7^−2/(−4)^−3
Since 15^0 is equal to 1, we can simplify it to:
1 ⋅ 7^−2/(−4)^−3
Next, let's apply the second property to divide the powers:
1 ⋅ 7^(-2-(-3))/(−4)^−3
Simplifying the exponent subtraction:
1 ⋅ 7^1/(−4)^−3
Now let's apply the third property to convert the negative exponent of (−4)^−3:
1 ⋅ 7^1/(−4)^3
Finally, let's simplify the expression:
1 ⋅ 7/(−4)^3
This gives us the expression 7/(−4)^3.
So, the correct option among the given choices is (−4)^3/7^2.