To determine whether Ramesh is correct about [tex]\( 7^{-5} \)[/tex], let's break down the calculation and understand the true value of [tex]\( 7^{-5} \)[/tex].
We are given the information that as the exponent decreases, the value is divided by 7. This pattern aligns with the rules of exponents for negative powers of a number.
Firstly, let's determine the true value of [tex]\( 7^{-5} \)[/tex]:
[tex]\[ 7^{-5} = \frac{1}{7^5} \][/tex]
Let's calculate [tex]\( 7^5 \)[/tex]:
[tex]\[ 7^5 = 7 \times 7 \times 7 \times 7 \times 7 = 16807 \][/tex]
Thus,
[tex]\[ 7^{-5} = \frac{1}{16807} \][/tex]
The numerical value of [tex]\(\frac{1}{16807}\)[/tex] can be approximated and it is:
[tex]\[ 7^{-5} \approx 5.9499018266198606 \times 10^{-5} \][/tex]
Now let's evaluate each statement provided in the question:
1. Ramesh is correct because [tex]\(7^{-5}\)[/tex] is equivalent to [tex]\(-7 \cdot (-7) \cdot (-7) \cdot (-7) \cdot (-7)\)[/tex], which has the same value as [tex]\(-16,807\)[/tex].
This statement is incorrect for several reasons:
- [tex]\( 7^{-5} \)[/tex] represents the reciprocal of [tex]\( 7^5 \)[/tex], not a series of multiplications involving negative sevens.
- [tex]\(-7 \cdot (-7) \cdot (-7) \cdot (-7) \cdot (-7)\)[/tex] would actually result in [tex]\(7^5 = 16807 \)[/tex] because multiplying an odd number of negative numbers still results in a negative number but not the correct context for reciprocal values.
2. Ramesh is correct because as the exponents decrease, the previous value is divided by 7, so [tex]\( 7^{-5} = 1 \div 7 \div 7 + 7 + 7 + 7 = -16,807 \)[/tex].
This statement is incorrect:
- While it is true that decreasing the exponent divides the previous value by 7, the arithmetic shown here mixes division and addition inappropriately to arrive at -16807, which is incorrect.
3. Ramesh is not correct because [tex]\( 7^{-5} \)[/tex] is equivalent to [tex]\(\frac{1}{7^5}\)[/tex], which has the same value as [tex]\(\frac{1}{7^4} \div 7 = \frac{1}{7^3} = \frac{1}{343}\)[/tex].
This statement is also incorrect:
- The correct relationship should be [tex]\( 7^{-5} = \frac{1}{7^5} \neq \frac{1}{7^4} \div 7 \)[/tex].
4. Ramesh is not correct because as the exponents decrease, the previous value is divided by 7, so [tex]\( 7^{-5} = 1 + 7 + 7 + 7 + 7 + 7 = \frac{1}{16807} \)[/tex].
This statement is part correct but contains errors:
- The reasoning behind decreased exponents involving division by 7 is correct, but the addition part is incorrect.
- However, it accurately specifies the correct final result [tex]\( \frac{1}{16807} \)[/tex].
Given the assessed calculations, the accurate statement should focus on the definition of [tex]\( 7^{-5} \)[/tex] as the reciprocal of [tex]\( 7^5 \)[/tex]:
Ramesh is not correct because [tex]\(7^{-5}\)[/tex] is equivalent to [tex]\(\frac{1}{7^5}\)[/tex], which has a value of approximately [tex]\(5.9499018266198606 \times 10^{-5}\)[/tex], not [tex]\(-16,807\)[/tex].
