Simplify the following expression:
[tex]\[ \frac{25^{\frac{2}{3}} \div 25^{\frac{1}{6}}}{\left(\frac{1}{5}\right)^{\frac{7}{6}} \times \left(\frac{1}{5}\right)^{\frac{1}{6}}} \][/tex]



Answer :

Let's solve the expression step-by-step:

[tex]\[ \frac{25^{\frac{2}{3}} \div 25^{\frac{1}{6}}}{\left(\frac{1}{5}\right)^{\frac{7}{6}} \times \left(\frac{1}{5}\right)^{\frac{1}{6}}} \][/tex]

### Step 1: Simplify the Numerator

First, let's consider the numerator [tex]\(25^{\frac{2}{3}} \div 25^{\frac{1}{6}}\)[/tex].

Using the property of exponents [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:

[tex]\[ 25^{\frac{2}{3}} \div 25^{\frac{1}{6}} = 25^{\frac{2}{3} - \frac{1}{6}} \][/tex]

We need to perform the subtraction [tex]\(\frac{2}{3} - \frac{1}{6}\)[/tex]:

Convert [tex]\(\frac{2}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex] to have a common denominator:
[tex]\[ \frac{2}{3} = \frac{4}{6} \][/tex]

Now, subtract the exponents:
[tex]\[ \frac{4}{6} - \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \][/tex]

So, the numerator simplifies to:
[tex]\[ 25^{\frac{1}{2}} \][/tex]

### Step 2: Simplify the Denominator

Next, let's simplify the denominator [tex]\(\left(\frac{1}{5}\right)^{\frac{7}{6}} \times \left(\frac{1}{5}\right)^{\frac{1}{6}}\)[/tex].

Using the property of exponents [tex]\(a^m \times a^n = a^{m+n}\)[/tex]:

[tex]\[ \left(\frac{1}{5}\right)^{\frac{7}{6}} \times \left(\frac{1}{5}\right)^{\frac{1}{6}} = \left(\frac{1}{5}\right)^{\frac{7}{6} + \frac{1}{6}} \][/tex]

Add the exponents:
[tex]\[ \frac{7}{6} + \frac{1}{6} = \frac{8}{6} = \frac{4}{3} \][/tex]

So, the denominator simplifies to:
[tex]\[ \left(\frac{1}{5}\right)^{\frac{4}{3}} \][/tex]

### Step 3: Combine the Results

Now, we have:
[tex]\[ \frac{25^{\frac{1}{2}}}{\left(\frac{1}{5}\right)^{\frac{4}{3}}} \][/tex]

Convert [tex]\(25^{\frac{1}{2}}\)[/tex] and [tex]\(\left(\frac{1}{5}\right)^{\frac{4}{3}}\)[/tex] to more familiar forms:
[tex]\[ 25^{\frac{1}{2}} = \sqrt{25} = 5 \][/tex]

[tex]\[ \left(\frac{1}{5}\right)^{\frac{4}{3}} = \left(5^{-1}\right)^{\frac{4}{3}} = 5^{-\frac{4}{3}} \][/tex]

So, the fraction becomes:
[tex]\[ \frac{5}{5^{-\frac{4}{3}}} \][/tex]

Using the property of exponents [tex]\(a^m \div a^n = a^{m-n}\)[/tex]:
[tex]\[ 5^{1 - \left(-\frac{4}{3}\right)} = 5^{1 + \frac{4}{3}} = 5^{\frac{3}{3} + \frac{4}{3}} = 5^{\frac{7}{3}} \][/tex]

### Step 4: Final Simplification

The final result is:
[tex]\[ 5^{\frac{7}{3}} \][/tex]

Converting [tex]\(5^{\frac{7}{3}}\)[/tex] to its numerical form:

[tex]\[ 5^{\frac{7}{3}} \approx 42.74939866691742 \][/tex]

Thus, the value of the given expression is approximately:

[tex]\[ 42.74939866691742 \][/tex]