Apply the product rule of integer exponents to find the numerical equivalent of 12−5⋅127 . Show your work and explain how you used the product rule of integer exponents.(2 points)
To apply the product rule of integer exponents to find the numerical equivalent of 12^(-5) * 127, we can use the property that a^(-n) = 1 / a^n.
First, let's simplify 12^(-5) using the product rule of integer exponents:
12^(-5) = (1 / 12^5)
Next, we can calculate the value of 12^5:
12^5 = 12 * 12 * 12 * 12 * 12 = 248,832
Now, substitute the value of 12^5 in the expression (1/12^5) * 127:
(1 / 12^5) * 127 = (1 / 248,832) * 127
Finally, we can calculate the numerical equivalent by dividing 127 by 248,832:
Numerical Equivalent = 127 / 248,832
This is the numerical value of 12^(-5) * 127, which can be calculated using the product rule of integer exponents.
To apply the product rule of integer exponents, we need to multiply the base while adding the exponents. In this case, we have:
12^(-5) * 127
The product rule states that for any base with integer exponents, we can multiply the bases and add the exponents.
Let's break down each factor:
1. 12^(-5)
To simplify this, we can rewrite it as the reciprocal of 12^5:
12^(-5) = 1 / 12^5
2. 127
There are no exponents involved, so we can leave it as it is.
Now, let's combine the two factors:
(1 / 12^5) * 127
To calculate this, we need to evaluate 12^5:
12^5 = 12 * 12 * 12 * 12 * 12 = 248,832
Now we can substitute this value back into the equation:
(1 / 248,832) * 127
To multiply fractions, we multiply the numerators and denominators separately:
1 * 127 = 127
248,832 * 1 = 248,832
So, the numerical equivalent of 12^(-5) * 127 is 127 / 248,832.
Note: I apologize that I made a calculation mistake and the answer is incorrect. The correct way to solve the equation is to evaluate 12^(-5) as the reciprocal of 12^5, resulting in 1/12^5. Then multiply this by 127. Hence, the correct answer is (1/12^5) * 127, which simplifies to 127/248,832.
To apply the product rule of integer exponents and find the numerical equivalent of 12^(-5) * 127, we need to start by understanding the rule itself. The product rule states that when multiplying two expressions with the same base but different exponents, you can add the exponents together while keeping the same base.
In this case, we have two factors: 12^(-5) and 127. Let's break it down step by step:
Step 1: Evaluate 12^(-5)
To evaluate 12^(-5), we know that raising a number to a negative exponent is equivalent to taking its reciprocal and raising it to the positive exponent. So, we can rewrite 12^(-5) as (1/12)^5.
Step 2: Calculate (1/12)^5
To calculate (1/12)^5, we raise the numerator and denominator separately to the power of 5. It becomes (1^5)/(12^5) = 1/12^5.
Step 3: Simplify 1/12^5
Simplifying 1/12^5, we need to evaluate the value of 12^5. It equals 12 * 12 * 12 * 12 * 12 = 248,832. Therefore, 1/12^5 becomes 1/248,832.
Now that we have simplified the first factor, let's continue with the multiplication:
Step 4: Multiply 1/248,832 by 127
To multiply 1/248,832 by 127, we multiply the numerators (1 * 127) and the denominators (248,832). This gives us the final result: 127/248,832.
Therefore, the numerical equivalent of 12^(-5) * 127 is 127/248,832.
In summary, we applied the product rule of integer exponents by treating the negative exponent as the reciprocal of the base raised to the positive exponent. By simplifying each factor separately and then performing the multiplication, we obtained the numerical equivalent of the given expression.