What is 5 to the power of 3 over five to the power of 7 in index form
The Correct Answer and Explanation is :
The expression “5 to the power of 3 over 5 to the power of 7” can be written as:
[
\frac{5^3}{5^7}
]
Step-by-Step Simplification:
In algebra, when dividing powers with the same base, you subtract the exponents. This is based on the law of exponents:
[
\frac{a^m}{a^n} = a^{m-n}
]
So, applying this rule:
[
\frac{5^3}{5^7} = 5^{3-7} = 5^{-4}
]
Thus, the expression simplifies to (5^{-4}).
Converting Negative Exponent:
A negative exponent can be rewritten as a reciprocal. That is:
[
a^{-n} = \frac{1}{a^n}
]
Therefore:
[
5^{-4} = \frac{1}{5^4}
]
Final Answer:
The final simplified form is:
[
\frac{1}{5^4}
]
Explanation (300 Words):
The original expression, (\frac{5^3}{5^7}), is a case of dividing powers with the same base, which simplifies by subtracting the exponents. This is a fundamental rule of exponents that states (\frac{a^m}{a^n} = a^{m-n}). In our case, both powers have a base of 5, and the exponents are 3 and 7. Subtracting the exponents (3 – 7) gives -4, so the expression simplifies to (5^{-4}).
A negative exponent indicates that the base (in this case, 5) should be placed in the denominator (i.e., made into a fraction). This is the result of another exponent rule: (a^{-n} = \frac{1}{a^n}). So, (5^{-4}) becomes (\frac{1}{5^4}).
To understand the practical meaning of this result, you could calculate (5^4), which equals 625. Therefore, the fraction (\frac{1}{5^4}) is the same as (\frac{1}{625}), representing a very small number.
This approach of simplifying powers with the same base makes it easy to handle large numbers and is commonly used in algebra, physics, and engineering to streamline calculations involving exponential terms.