Using the Quotient Rule of Exponents, what is the exponent in an equivalent form of 7976 if the base is 7
The Quotient Rule of Exponents states that when dividing two numbers with the same base, you subtract the exponents.
In this case, we have 7976 as the base raised to some exponent. We want to find the exponent in an equivalent form when the base is 7.
Let's say the exponent we are looking for is "x". Using the Quotient Rule of Exponents, we can set up the following equation:
7^x = 7976
To solve for x, we need to take the logarithm of both sides of the equation. Using the natural logarithm (ln) as an example, we have:
ln(7^x) = ln(7976)
Using the logarithmic property that ln(a^b) = b * ln(a), we can simplify the left side:
x * ln(7) = ln(7976)
Now, to solve for x, we divide both sides of the equation by ln(7):
x = ln(7976) / ln(7)
Using a calculator, we can find the approximate value of x to be approximately 5.439.
Therefore, the exponent in an equivalent form of 7976 when the base is 7 is approximately 5.439.
To use the Quotient Rule of Exponents, we need to divide the exponents when dividing two powers of the same base.
The Quotient Rule of Exponents states that for any positive numbers a and b, and any nonzero number m:
a^m / a^n = a^(m-n)
In this case, we are given a base of 7. Therefore, if we have 7^m / 7^n, we can simplify it to 7^(m-n).
To find the exponent in an equivalent form of 7976, we need to break it down as follows:
7976 = 7^m / 7^n
Since the base is the same (7), we can simplify it as:
7^(m - n) = 7976
Now we need to consider the prime factorization of 7976.
The prime factorization of 7976 is:
7976 = 2^3 * 997
Therefore, we can rewrite the equation as:
7^(m - n) = 2^3 * 997
Now, since 7 and 2 are prime numbers and they cannot be further simplified, the exponents must be equal on both sides of the equation.
So, we can set up the following equation:
m - n = 3
This tells us that the exponent in an equivalent form of 7976, with a base of 7, is 3.
The Quotient Rule of Exponents states that when dividing two numbers with the same base, you subtract the exponents.
In this case, we have the number 7976 with a base of 7. We want to find the exponent in an equivalent form.
To find the exponent, we need to rewrite 7976 as a division of powers of 7. Let's start by factoring 7976.
7976 = 2 * 3988
Now, let's continue factoring:
3988 = 2 * 1994
1994 = 2 * 997
997 is a prime number, so we stop here.
Now, let's rewrite 7976 as a division of powers of 7:
7976 = 2 * 3988
= 2 * (2 * 1994)
= 2 * (2 * (2 * 997))
We can see that 7976 can be written as 2^3 * 997.
Now, let's rewrite 2^3 * 997 using the Quotient Rule of Exponents:
2^3 * 997 = 7^(3 - 0) * 997
Therefore, the exponent in an equivalent form of 7976 with a base of 7 is 3.