Let's solve the given expression step by step:
[tex]\[
\frac{\left(\frac{1}{4}\right)^{15} \div \left(\frac{1}{4}\right)^5}{\left(\frac{1}{7}\right)^{10} \left(\frac{1}{4}\right)^5}
\][/tex]
### Step 1: Simplify the Numerator
First, simplify the numerator [tex]\(\left(\frac{1}{4}\right)^{15} \div \left(\frac{1}{4}\right)^5\)[/tex]:
We can use the property of exponents [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[
\left(\frac{1}{4}\right)^{15} \div \left(\frac{1}{4}\right)^5 = \left(\frac{1}{4}\right)^{15-5} = \left(\frac{1}{4}\right)^{10}
\][/tex]
So the simplified numerator is:
[tex]\[
\left(\frac{1}{4}\right)^{10}
\][/tex]
### Step 2: Simplify the Denominator
Next, simplify the denominator [tex]\(\left(\frac{1}{7}\right)^{10} \times \left(\frac{1}{4}\right)^5\)[/tex]:
Leave this as is since it is already in its simplest form.
### Step 3: Combine the Simplified Numerator and Denominator
We now need to express the whole expression as:
[tex]\[
\frac{\left(\frac{1}{4}\right)^{10}}{\left(\frac{1}{7}\right)^{10} \left(\frac{1}{4}\right)^5}
\][/tex]
### Step 4: Simplify the Expression
We can rewrite the expression using the properties of exponents:
[tex]\[
\frac{\left(\frac{1}{4}\right)^{10}}{\left(\frac{1}{7}\right)^{10} \left(\frac{1}{4}\right)^5} = \frac{\left(\frac{1}{4}\right)^{10}}{\left(\frac{1}{4}\right)^5 \left(\frac{1}{7}\right)^{10}}
\][/tex]
Using the property of exponents again, [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[
\frac{\left(\frac{1}{4}\right)^{10}}{\left(\frac{1}{4}\right)^5} = \left(\frac{1}{4}\right)^{10-5} = \left(\frac{1}{4}\right)^{5}
\][/tex]
So the expression now becomes:
[tex]\[
\frac{\left(\frac{1}{4}\right)^{5}}{\left(\frac{1}{7}\right)^{10}}
\][/tex]
### Step 5: Express as a Single Fraction
[tex]\(\left(\frac{1}{4}\right)^{5}\)[/tex] and [tex]\(\left(\frac{1}{7}\right)^{10}\)[/tex] can be written as:
[tex]\[
\frac{\left(\frac{1}{4}\right)^{5}}{\left(\frac{1}{7}\right)^{10}} = \frac{1}{4^5} \times \frac{7^{10}}{1}
\][/tex]
Combine these:
[tex]\[
\frac{7^{10}}{4^5}
\][/tex]
### Step 6: Calculate the Final Result
To obtain the precise numerical values:
[tex]\[
4^5 = 1024 \quad \text{and} \quad 7^{10} = 282475249
\][/tex]
So the expression becomes:
[tex]\[
\frac{282475249}{1024}
\][/tex]
Dividing:
[tex]\[
\frac{282475249}{1024} \approx 275854.7353515626
\][/tex]
Therefore, the final result of the given expression is:
[tex]\[
\boxed{275854.7353515626}
\][/tex]
