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?
explain how the answer is (−4)^3/7^2
To simplify the expression 15^0 ⋅ 7^−2 (−4)^−3, we can start by evaluating each term individually.
First, 15^0 is equal to 1. Any number raised to the power of 0 is always equal to 1.
Next, 7^−2 can be written as 1/7^2. A negative exponent indicates that the base should be moved to the denominator and the sign of the exponent should change. So 7^−2 is equivalent to 1/7^2.
Then, (−4)^−3 can be written as 1/(-4)^3. Here, the negative exponent similarly indicates that the base should be moved to the denominator and the sign of the exponent should change. So (−4)^−3 is equivalent to 1/(-4)^3.
Now, let's combine these individual expressions: 1 ⋅ 1/7^2 ⋅ 1/(-4)^3.
To simplify further, we can rewrite 1/7^2 as 7^−2.
So the expression becomes 1 ⋅ 7^−2 ⋅ 1/(-4)^3.
Next, we can rearrange the expression by placing the negative exponent before the base. So it becomes 1 ⋅ 7^−2 ⋅ (-4)^3.
Finally, applying the property of multiplying exponents with the same base, we can rewrite 7^−2 ⋅ (-4)^3 as (-4)^3/7^2.
Therefore, the equivalent expression to 15^0 ⋅ 7^−2 (−4)^−3 with only positive exponents is (-4)^3/7^2.
To simplify the expression and convert the exponents to positive, we can apply the following property of integer exponents:
(a^m)^n = a^(m*n)
Using this property, let's simplify each part of the original expression step-by-step:
Step 1: Simplify 15^0
Any number (except zero) raised to the power of 0 is equal to 1. Therefore, 15^0 = 1.
Step 2: Simplify 7^(-2)
To convert a negative exponent to a positive exponent, we can use the following property:
a^(-n) = 1 / a^n
Applying this property to 7^(-2):
7^(-2) = 1 / 7^2
Step 3: Simplify (-4)^(-3)
Similar to Step 2, we can apply the property:
a^(-n) = 1 / a^n
So, (-4)^(-3) = 1 / (-4)^3
Step 4: Combining all the simplified terms
Combining the simplified terms from Steps 1, 2, and 3:
15^0 ⋅ 7^(-2) ⋅ (-4)^(-3) = 1 ⋅ (1 / 7^2) ⋅ (1 / (-4)^3)
Since we want to express the equivalent expression with only positive exponents, we can simply write this as:
(1 / (-4)^3) / 7^2
Which can be further simplified as:
(-4)^3 / 7^2
Therefore, the equivalent expression with only positive exponents that has been generated by applying the Properties of Integer Exponents is (-4)^3 / 7^2.
To simplify the given expression and generate an equivalent expression with only positive exponents, we can apply the properties of integer exponents.
Let's break down the expression step by step:
Step 1: Simplify 15^0
Any number raised to the power of 0 is equal to 1. Therefore, 15^0 is equal to 1.
Step 2: Simplify 7^−2
To simplify a negative exponent, we can take its reciprocal with a positive exponent. In this case, 7^−2 is equivalent to 1/7^2.
Step 3: Simplify (−4)^−3
Similar to the previous step, we can rewrite a negative exponent as the reciprocal of its positive exponent. Thus, (−4)^−3 is equal to 1/(−4)^3.
Step 4: Combine the simplified expressions from steps 1, 2, and 3
Taking the results from the previous steps, we have 15^0 ⋅ 7^−2 ⋅ (−4)^−3 = 1 ⋅ 1/7^2 ⋅ 1/(−4)^3.
Step 5: Apply the property of exponents
The property of exponents states that when multiplying two expressions with the same base, we add their exponents. Based on this property, we can rewrite the expression as 1/7^2 ⋅ 1/(−4)^3 = (1/7^2) ⋅ (1/(−4)^3).
Step 6: Simplify the exponents
To simplify the expression further, we can calculate the values of the exponents. 7^2 is equal to 49, and (−4)^3 is equal to -64.
Final step: Put it all together
Now, we can substitute the calculated values into our expression to obtain the final answer: (1/49) ⋅ (1/(-64)). Multiplying the two terms gives us 1/(-3136).
Therefore, an equivalent expression to 15^0 ⋅ 7^−2 ⋅ (−4)^−3, with only positive exponents, is (−4)^3 / 7^2, which simplifies to 1/(-3136).