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. A 30 A^{30} A 30 C.

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Introduction

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Exponential expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for any math enthusiast. In this article, we will explore the simplified form of the expression $\left(\left(p^2\right)\left(q^5\right)\right)^{-4} \cdot\left(\left(p^{-4}\right)\left(q^5\right)\right)^{-2}$, and we will break down the steps involved in simplifying it.

Understanding Exponents

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Before we dive into the simplification process, let's review the basics of exponents. An exponent is a small number that is written to the upper right of a number or a variable. It represents the power to which the base is raised. For example, in the expression $a^3$, the exponent 3 represents the power to which the base $a$ is raised.

Simplifying the Expression

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Now that we have a basic understanding of exponents, let's simplify the given expression. The expression is $\left(\left(p^2\right)\left(q^5\right)\right)^{-4} \cdot\left(\left(p^{-4}\right)\left(q^5\right)\right)^{-2}$.

Step 1: Apply the Power Rule

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The power rule states that for any numbers $a$ and $b$ and any integers $m$ and $n$, $(ab)^m = a^mb^m$. We can apply this rule to the given expression by distributing the exponent $-4$ to both $p^2$ and $q^5$.

$\left(\left(p^2\right)\left(q^5\right)\right)^{-4} = \left(p^2\right)^{-4}\left(q^5\right)^{-4} = p^{-8}q^{-20}$

Similarly, we can apply the power rule to the second part of the expression.

$\left(\left(p^{-4}\right)\left(q^5\right)\right)^{-2} = \left(p^{-4}\right)^{-2}\left(q^5\right)^{-2} = p^8q^{-10}$

Step 2: Multiply the Expressions

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Now that we have simplified both parts of the expression, we can multiply them together.

$p^{-8}q^{-20} \cdot p^8q^{-10} = p^{-8+8}q^{-20-10} = p^0q^{-30}$

Step 3: Simplify the Expression

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Finally, we can simplify the expression by applying the rule that $a^0 = 1$ for any nonzero number $a$.

$p^0q^{-30} = 1 \cdot q^{-30} = \frac{1}{q^{30}}$

Conclusion

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In conclusion, 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 $\frac{1}{q^{30}}$. We applied the power rule to simplify the expression and then multiplied the two parts together. Finally, we simplified the expression by applying the rule that $a^0 = 1$ for any nonzero number $a$.

Frequently Asked Questions

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Q: What is the power rule in mathematics?

A: The power rule is a rule in mathematics that states that for any numbers $a$ and $b$ and any integers $m$ and $n$, $(ab)^m = a^mb^m$.

Q: How do I simplify an exponential expression?

A: To simplify an exponential expression, you can apply the power rule by distributing the exponent to both the base and the exponent. Then, you can multiply the two parts of the expression together and simplify the result.

Q: What is the rule for $a^0$?

A: The rule for $a^0$ is that $a^0 = 1$ for any nonzero number $a$.

Final Answer

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The final answer is $\boxed{\frac{1}{q^{30}}}$.

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Introduction

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Exponential expressions are a fundamental concept in mathematics, and understanding them is crucial for success in math and science. In this article, we will answer some of the most frequently asked questions about exponential expressions, covering topics such as simplifying expressions, applying the power rule, and more.

Q&A

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Q: What is the power rule in mathematics?

A: The power rule is a rule in mathematics that states that for any numbers $a$ and $b$ and any integers $m$ and $n$, $(ab)^m = a^mb^m$. This rule allows us to simplify complex exponential expressions by distributing the exponent to both the base and the exponent.

Q: How do I simplify an exponential expression?

A: To simplify an exponential expression, you can apply the power rule by distributing the exponent to both the base and the exponent. Then, you can multiply the two parts of the expression together and simplify the result.

Q: What is the rule for $a^0$?

A: The rule for $a^0$ is that $a^0 = 1$ for any nonzero number $a$. This means that any number raised to the power of 0 is equal to 1.

Q: How do I handle negative exponents?

A: Negative exponents can be handled by applying the rule that $a^{-n} = \frac{1}{a^n}$. This means that any number raised to a negative power is equal to the reciprocal of the number raised to the positive power.

Q: Can I simplify an expression with multiple bases?

A: Yes, you can simplify an expression with multiple bases by applying the power rule to each base separately. For example, if you have the expression $(ab)^m$, you can simplify it by applying the power rule to both $a$ and $b$ separately.

Q: How do I handle exponents with fractions?

A: Exponents with fractions can be handled by applying the rule that $(a^m)^n = a^{mn}$. This means that any number raised to a power that is itself a fraction can be simplified by multiplying the exponents.

Q: Can I simplify an expression with a zero exponent?

A: Yes, you can simplify an expression with a zero exponent by applying the rule that $a^0 = 1$ for any nonzero number $a$. This means that any number raised to the power of 0 is equal to 1.

Examples

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Example 1: Simplifying an Expression with Multiple Bases

Suppose we have the expression $(ab)^m$. We can simplify this expression by applying the power rule to both $a$ and $b$ separately.

$(ab)^m = a^mb^m$

Example 2: Handling Negative Exponents

Suppose we have the expression $a^{-n}$. We can simplify this expression by applying the rule that $a^{-n} = \frac{1}{a^n}$.

$a^{-n} = \frac{1}{a^n}$

Example 3: Simplifying an Expression with a Zero Exponent

Suppose we have the expression $a^0$. We can simplify this expression by applying the rule that $a^0 = 1$ for any nonzero number $a$.

$a^0 = 1$

Conclusion

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In conclusion, exponential expressions are a fundamental concept in mathematics, and understanding them is crucial for success in math and science. By applying the power rule, handling negative exponents, and simplifying expressions with multiple bases, you can simplify complex exponential expressions and solve a wide range of problems.

Frequently Asked Questions

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Q: What is the power rule in mathematics?

A: The power rule is a rule in mathematics that states that for any numbers $a$ and $b$ and any integers $m$ and $n$, $(ab)^m = a^mb^m$.

Q: How do I simplify an exponential expression?

A: To simplify an exponential expression, you can apply the power rule by distributing the exponent to both the base and the exponent. Then, you can multiply the two parts of the expression together and simplify the result.

Q: What is the rule for $a^0$?

A: The rule for $a^0$ is that $a^0 = 1$ for any nonzero number $a$.

Final Answer

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The final answer is $\boxed{\frac{1}{q^{30}}}$.