Simplify The Expression:$\left((-3)^3 S^5\right)^2 - R^6 S^{10}$

Introduction

In this article, we will simplify the given expression: $\left((-3)^3 s^5\right)^2 - r^6 s^{10}$. This involves applying the rules of exponents and simplifying the resulting expression. We will break down the process into manageable steps, making it easier to understand and follow along.

Step 1: Apply the Power of a Power Rule

The first step is to apply the power of a power rule, which states that for any numbers $a$ and $b$ and any integers $m$ and $n$, $(a^m)^n = a^{m \cdot n}$. In this case, we have $\left((-3)^3 s^5\right)^2$. Applying the power of a power rule, we get:

$((-3)^3 s^5)^2 = (-3)^{3 \cdot 2} s^{5 \cdot 2} = (-3)^6 s^{10}$

Step 2: Simplify the Expression

Now that we have simplified the first part of the expression, we can rewrite the original expression as:

$(-3)^6 s^{10} - r^6 s^{10}$

Step 3: Factor Out the Common Term

The next step is to factor out the common term, which is $s^{10}$. We can do this by recognizing that both terms have a factor of $s^{10}$ in common. Factoring out the common term, we get:

$s^{10}((-3)^6 - r^6)$

Step 4: Simplify the Expression Inside the Parentheses

Now that we have factored out the common term, we can simplify the expression inside the parentheses. We have $(-3)^6 - r^6$. To simplify this expression, we can use the fact that $(-3)^6 = 729$ and $r^6$ is just $r^6$. Therefore, we can rewrite the expression as:

$s^{10}(729 - r^6)$

Step 5: Final Simplification

The final step is to simplify the expression by combining the terms. We have $s^{10}(729 - r^6)$. This is the final simplified expression.

Conclusion

In this article, we simplified the given expression: $\left((-3)^3 s^5\right)^2 - r^6 s^{10}$. We applied the power of a power rule, simplified the expression, factored out the common term, and finally simplified the expression inside the parentheses. The final simplified expression is $s^{10}(729 - r^6)$.

Key Takeaways

• The power of a power rule states that for any numbers $a$ and $b$ and any integers $m$ and $n$, $(a^m)^n = a^{m \cdot n}$.
• Factoring out the common term can help simplify complex expressions.
• Simplifying the expression inside the parentheses can help reveal the final simplified expression.

Real-World Applications

Simplifying expressions is an essential skill in mathematics, and it has many real-world applications. For example, in physics, simplifying expressions can help us understand complex phenomena, such as the motion of objects. In engineering, simplifying expressions can help us design and optimize systems. In finance, simplifying expressions can help us understand complex financial models.

Common Mistakes to Avoid

When simplifying expressions, it's essential to avoid common mistakes. Some common mistakes include:

• Not applying the power of a power rule correctly.
• Not factoring out the common term.
• Not simplifying the expression inside the parentheses.

Final Thoughts

Introduction

In our previous article, we simplified the expression: $\left((-3)^3 s^5\right)^2 - r^6 s^{10}$. We applied the power of a power rule, simplified the expression, factored out the common term, and finally simplified the expression inside the parentheses. In this article, we will answer some frequently asked questions about simplifying expressions.

Q: What is the power of a power rule?

A: The power of a power rule states that for any numbers $a$ and $b$ and any integers $m$ and $n$, $(a^m)^n = a^{m \cdot n}$. This rule allows us to simplify expressions by combining the exponents.

Q: How do I apply the power of a power rule?

A: To apply the power of a power rule, simply multiply the exponents. For example, if we have $(a^3)^2$, we can apply the power of a power rule by multiplying the exponents: $(a^3)^2 = a^{3 \cdot 2} = a^6$.

Q: What is factoring out the common term?

A: Factoring out the common term is a technique used to simplify expressions by identifying a common factor and factoring it out. For example, if we have $2x + 4x$, we can factor out the common term $x$ by writing it as $x(2 + 4)$.

Q: How do I simplify the expression inside the parentheses?

A: To simplify the expression inside the parentheses, we need to evaluate the expression inside the parentheses. For example, if we have $(2 + 3)$, we can simplify it by evaluating the expression inside the parentheses: $(2 + 3) = 5$.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

• Not applying the power of a power rule correctly.
• Not factoring out the common term.
• Not simplifying the expression inside the parentheses.
• Not combining like terms.

Q: How do I know when to simplify an expression?

A: We should simplify an expression when it is necessary to make the expression easier to understand or to evaluate. For example, if we have a complex expression that involves multiple operations, we may need to simplify it to make it easier to evaluate.

Q: Can I simplify expressions with variables?

A: Yes, we can simplify expressions with variables. In fact, simplifying expressions with variables is an essential skill in algebra. We can use the same techniques we use to simplify expressions with numbers to simplify expressions with variables.

Q: How do I know when to use the distributive property?

A: We should use the distributive property when we need to multiply a single term by multiple terms. For example, if we have $a(b + c)$, we can use the distributive property to write it as $ab + ac$.

Q: Can I simplify expressions with fractions?

A: Yes, we can simplify expressions with fractions. In fact, simplifying expressions with fractions is an essential skill in algebra. We can use the same techniques we use to simplify expressions with numbers to simplify expressions with fractions.

Conclusion

In this article, we answered some frequently asked questions about simplifying expressions. We discussed the power of a power rule, factoring out the common term, simplifying the expression inside the parentheses, and common mistakes to avoid. We also discussed how to simplify expressions with variables, fractions, and when to use the distributive property. By following the tips and techniques outlined in this article, you can become proficient in simplifying expressions and tackle complex mathematical problems with confidence.