Sure! Let's solve the given expression step-by-step:
The expression we need to evaluate is:
[tex]\[
\frac{\left(5^{-7}\right)^3}{5^4 \cdot 5^7}
\][/tex]
### Step 1: Simplify the Numerator
First, we apply the power of a power rule in the numerator [tex]\(\left( a^m \right)^n = a^{m \cdot n} \)[/tex]:
[tex]\[
(5^{-7})^3 = 5^{-7 \cdot 3} = 5^{-21}
\][/tex]
So, the expression now looks like this:
[tex]\[
\frac{5^{-21}}{5^4 \cdot 5^7}
\][/tex]
### Step 2: Simplify the Denominator
Next, we apply the product of powers rule in the denominator [tex]\(a^m \cdot a^n = a^{m + n}\)[/tex]:
[tex]\[
5^4 \cdot 5^7 = 5^{4 + 7} = 5^{11}
\][/tex]
So, the expression now looks like this:
[tex]\[
\frac{5^{-21}}{5^{11}}
\][/tex]
### Step 3: Apply the Quotient of Powers Rule
Finally, we apply the quotient of powers rule [tex]\(a^m / a^n = a^{m - n}\)[/tex]:
[tex]\[
\frac{5^{-21}}{5^{11}} = 5^{-21 - 11} = 5^{-32}
\][/tex]
### Step 4: Compute the Final Result
So, the simplified form of the expression is:
[tex]\[
5^{-32}
\][/tex]
To provide the numerical value:
[tex]\[
5^{-32} \approx 4.294967296 \times 10^{-23}
\][/tex]
### Summary of Results
- The numerator, [tex]\(5^{-21}\)[/tex], when evaluated is approximately [tex]\(2.097152 \times 10^{-15}\)[/tex].
- The denominator, [tex]\(5^{11}\)[/tex], when evaluated is [tex]\(48828125\)[/tex].
- The final result, [tex]\(5^{-32}\)[/tex], is approximately [tex]\(4.294967296 \times 10^{-23}\)[/tex].
Therefore, the exact expression simplifies to [tex]\(5^{-32}\)[/tex], which is approximately [tex]\(4.294967296 \times 10^{-23}\)[/tex] when evaluated numerically.
