Which Is The Simplified Form Of The Expression ( ( P 2 ) ( Q 5 ) ) − 4 ⋅ ( ( P − 4 ) ( Q 5 ) ) − 2 \left(\left(p^2\right)\left(q^5\right)\right)^{-4} \cdot\left(\left(p^{-4}\right)\left(q^5\right)\right)^{-2} ( ( P 2 ) ( Q 5 ) ) − 4 ⋅ ( ( P − 4 ) ( Q 5 ) ) − 2 ?A. 1 Q 30 \frac{1}{q^{30}} Q 30 1 ​ B. Q 30 Q^{30} Q 30 C.

**Simplified Form of Algebraic Expressions: A Step-by-Step Guide** ===========================================================

Understanding the Problem

The given problem involves simplifying an algebraic expression that contains exponents and variables. The expression is $\left(\left(p^2\right)\left(q^5\right)\right)^{-4} \cdot\left(\left(p^{-4}\right)\left(q^5\right)\right)^{-2}$, and we need to find its simplified form.

Step 1: Apply the Power Rule for Exponents

To simplify the given expression, we need to apply the power rule for exponents, which states that for any variables $a$ and $b$ and any integers $m$ and $n$, $(a^m)^n = a^{m \cdot n}$.

Using this rule, we can rewrite the given expression as:

$\left(p^2\right)^{-4} \cdot \left(q^5\right)^{-4} \cdot \left(p^{-4}\right)^{-2} \cdot \left(q^5\right)^{-2}$

Step 2: Simplify the Exponents

Now, we can simplify the exponents by multiplying the exponents of the same base:

$p^{2 \cdot (-4)} \cdot q^{5 \cdot (-4)} \cdot p^{-4 \cdot (-2)} \cdot q^{5 \cdot (-2)}$

This simplifies to:

$p^{-8} \cdot q^{-20} \cdot p^{8} \cdot q^{-10}$

Step 3: Combine Like Terms

Next, we can combine like terms by adding or subtracting the exponents of the same base:

$p^{-8 + 8} \cdot q^{-20 - 10}$

This simplifies to:

$p^0 \cdot q^{-30}$

Step 4: Simplify the Expression

Finally, we can simplify the expression by applying the rule that $a^0 = 1$ for any non-zero value of $a$:

$p^0 = 1$

Therefore, the simplified form of the expression is:

$\boxed{q^{-30}}$

Answer:

The simplified form of the expression $\left(\left(p^2\right)\left(q^5\right)\right)^{-4} \cdot\left(\left(p^{-4}\right)\left(q^5\right)\right)^{-2}$ is $\boxed{q^{-30}}$.

Q&A

Q: What is the power rule for exponents?

A: The power rule for exponents states that for any variables $a$ and $b$ and any integers $m$ and $n$, $(a^m)^n = a^{m \cdot n}$.

Q: How do you simplify exponents using the power rule?

A: To simplify exponents using the power rule, you multiply the exponents of the same base.

Q: What is the rule for combining like terms?

A: The rule for combining like terms is to add or subtract the exponents of the same base.

Q: What is the simplified form of the expression $\left(\left(p^2\right)\left(q^5\right)\right)^{-4} \cdot\left(\left(p^{-4}\right)\left(q^5\right)\right)^{-2}$?

A: The simplified form of the expression is $\boxed{q^{-30}}$.

Q: What is the value of $p^0$?

A: The value of $p^0$ is 1 for any non-zero value of $p$.

Q: What is the value of $q^{-30}$?

A: The value of $q^{-30}$ is $\frac{1}{q^{30}}$.