Certainly! We need to simplify the given expression step by step:
[tex]$\frac{\left(5^{-7}\right)^3}{5^4 \cdot 5^7}$[/tex]
1. Simplify the numerator:
- The numerator is [tex]\(\left(5^{-7}\right)^3\)[/tex].
- When raising a power to another power, we multiply the exponents:
[tex]$ (5^{-7})^3 = 5^{-7 \cdot 3} = 5^{-21}$[/tex]
2. Simplify the denominator:
- The denominator is [tex]\(5^4 \cdot 5^7\)[/tex].
- When multiplying like bases, we add the exponents:
[tex]$ 5^4 \cdot 5^7 = 5^{4 + 7} = 5^{11} $[/tex]
3. Rewrite the expression with the simplified numerator and denominator:
[tex]$ \frac{5^{-21}}{5^{11}} $[/tex]
4. Simplify the fraction:
- When dividing like bases, we subtract the exponents (numerator exponent minus denominator exponent):
[tex]$ \frac{5^{-21}}{5^{11}} = 5^{-21 - 11} = 5^{-32} $[/tex]
5. Evaluate the expression:
- To find the value of [tex]\(5^{-32}\)[/tex], we recognize that negative exponents represent the reciprocal:
[tex]$ 5^{-32} = \frac{1}{5^{32}} $[/tex]
6. Calculate the value:
- Since [tex]\(5^{32}\)[/tex] is a very large number, its reciprocal [tex]\(5^{-32}\)[/tex] is a very small number:
[tex]$ 5^{-32} = 4.294967296 \times 10^{-23} $[/tex]
Hence, the final value of the given expression is:
[tex]$\frac{\left(5^{-7}\right)^3}{5^4 \cdot 5^7} = 5^{-32} = 4.294967296 \times 10^{-23} $[/tex]
