Simplify ((1/625)^2022) x ((5^2021)^5) x ((1/25)^2022) x ((25^2021)^1/2)

Let's simplify the given expression step by step:

\[
\left(\frac{1}{625}\right)^{2022} \times (5^{2021})^5 \times \left(\frac{1}{25}\right)^{2022} \times (25^{2021})^{\frac{1}{2}}
\]

### Step 1: Express powers of base numbers in terms of powers of 5
We know the following relationships:
\[
625 = 5^4 \quad \text{and} \quad 25 = 5^2
\]
Now rewrite the expression with these substitutions:
\[
\left(\frac{1}{5^4}\right)^{2022} \times (5^{2021})^5 \times \left(\frac{1}{5^2}\right)^{2022} \times (5^2)^{2021 \times \frac{1}{2}}
\]

### Step 2: Simplify the exponents
Now simplify each term:
1. \(\left(\frac{1}{5^4}\right)^{2022} = 5^{-4 \times 2022} = 5^{-8088}\)
2. \((5^{2021})^5 = 5^{2021 \times 5} = 5^{10105}\)
3. \(\left(\frac{1}{5^2}\right)^{2022} = 5^{-2 \times 2022} = 5^{-4044}\)
4. \((5^2)^{2021 \times \frac{1}{2}} = 5^{2 \times \frac{2021}{2}} = 5^{2021}\)

Now the expression becomes:
\[
5^{-8088} \times 5^{10105} \times 5^{-4044} \times 5^{2021}
\]

### Step 3: Combine the exponents
Since the base is the same (all terms involve \(5^x\)), we can combine the exponents:
\[
5^{-8088 + 10105 - 4044 + 2021}
\]
Now simplify the exponents:
\[
-8088 + 10105 - 4044 + 2021 = -8088 + 10105 = 2017, \quad 2017 - 4044 = -2027, \quad -2027 + 2021 = -6
\]

Thus, the expression simplifies to:
\[
5^{-6}
\]

This can also be written as:
\[
\frac{1}{5^6} = \frac{1}{15625}
\]

### Final Answer:
The simplified form of the expression is:
\[
\frac{1}{15625}
\]

Would you like more details on any specific part of the solution?

Here are 5 related questions for practice:
1. Simplify \((5^3)^4 \times (5^2)^6\).
2. How would you simplify \((1/125)^3 \times (25/5)^2\)?
3. What is the simplified result of \((5^{10}) / (5^4 \times 5^3)\)?
4. Simplify \(\left(\frac{1}{5^3}\right)^4 \times (5^2)^{-2}\).
5. If \(a = 5^2\) and \(b = 5^3\), express \(a^4 \times b^3\) in terms of powers of 5.

**Tip:** When simplifying expressions with powers of the same base, always combine exponents by adding or subtracting them depending on multiplication or division.

This detailed solution explains how to simplify the expression ((1/625)^2022) x ((5^2021)^5) x ((1/25)^2022) x ((25^2021)^1/2) using the laws of exponents and powers. Learn key mathematical concepts such as exponent rules, simplifying terms, and combining powers. Suitable for students in Grades 10-12.

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