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

Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and manipulate mathematical statements. In this article, we will focus on simplifying the given expression: $\left(-3r^3 s^5\right)^2 - r^6 s^{10}$. We will break down the problem into manageable steps, using algebraic properties and rules to simplify the expression.

Understanding the Expression

The given expression is a difference of two terms, each involving powers of $r$ and $s$. The first term is $\left(-3r^3 s^5\right)^2$, which means we need to square the entire expression inside the parentheses. The second term is $r^6 s^{10}$, which is a straightforward term.

Step 1: Square the First Term

To square the first term, we need to apply the exponent rule $\left(ab\right)^n = a^n b^n$. In this case, we have $\left(-3r^3 s^5\right)^2$, which means we need to square both the coefficient $-3$ and the variables $r^3$ and $s^5$.

(-3r^3 s^5)^2 = (-3)^2 (r^3)^2 (s^5)^2

Using the exponent rule, we can simplify the expression as follows:

(-3)^2 = 9
(r^3)^2 = r^6
(s^5)^2 = s^{10}

So, the squared first term becomes:

9r^6 s^{10}

Step 2: Subtract the Second Term

Now that we have simplified the first term, we can subtract the second term $r^6 s^{10}$ from it.

9r^6 s^{10} - r^6 s^{10}

Using the distributive property, we can combine like terms:

(9 - 1)r^6 s^{10}

Simplifying further, we get:

8r^6 s^{10}

Conclusion

In this article, we simplified the given expression $\left(-3r^3 s^5\right)^2 - r^6 s^{10}$ using algebraic properties and rules. We broke down the problem into manageable steps, squaring the first term and subtracting the second term. The final simplified expression is $8r^6 s^{10}$.

Tips and Tricks

• When simplifying expressions, always look for opportunities to combine like terms.
• Use exponent rules to simplify expressions involving powers of variables.
• Apply the distributive property to combine like terms.

Common Mistakes

• Failing to simplify expressions by combining like terms.
• Not applying exponent rules to simplify expressions involving powers of variables.
• Not using the distributive property to combine like terms.

Real-World Applications

Simplifying expressions is a crucial skill in many real-world applications, including:

• Physics: Simplifying expressions is essential in physics to solve equations and manipulate mathematical statements.
• Engineering: Simplifying expressions is critical in engineering to design and analyze systems.
• Computer Science: Simplifying expressions is essential in computer science to optimize algorithms and data structures.

Final Thoughts

Introduction

In our previous article, we simplified the expression $\left(-3r^3 s^5\right)^2 - r^6 s^{10}$ using algebraic properties and rules. In this article, we will answer some frequently asked questions (FAQs) related to simplifying expressions.

Q&A

Q: What is the difference between simplifying an expression and solving an equation?

A: Simplifying an expression involves reducing it to its simplest form, while solving an equation involves finding the value of the variable that makes the equation true.

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

A: You should simplify an expression whenever it appears in a mathematical statement or equation. Simplifying expressions helps to make the statement or equation easier to understand and work with.

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

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

• Failing to simplify expressions by combining like terms.
• Not applying exponent rules to simplify expressions involving powers of variables.
• Not using the distributive property to combine like terms.
• Not checking for errors in the simplification process.

Q: How do I simplify expressions involving fractions?

A: To simplify expressions involving fractions, you can use the following steps:

1. Simplify the numerator and denominator separately.
2. Look for common factors between the numerator and denominator.
3. Cancel out any common factors.

Q: How do I simplify expressions involving exponents?

A: To simplify expressions involving exponents, you can use the following steps:

1. Simplify the base and exponent separately.
2. Apply the exponent rule $\left(ab\right)^n = a^n b^n$.
3. Simplify the resulting expression.

Q: What are some real-world applications of simplifying expressions?

A: Simplifying expressions has many real-world applications, including:

• Physics: Simplifying expressions is essential in physics to solve equations and manipulate mathematical statements.
• Engineering: Simplifying expressions is critical in engineering to design and analyze systems.
• Computer Science: Simplifying expressions is essential in computer science to optimize algorithms and data structures.

Q: How do I check my work when simplifying expressions?

A: To check your work when simplifying expressions, you can use the following steps:

1. Plug in a value for the variable and simplify the expression.
2. Check that the simplified expression is equal to the original expression.
3. Verify that the simplified expression is correct.

Tips and Tricks

• Always check your work when simplifying expressions.
• Use a calculator or computer program to check your work if necessary.
• Simplify expressions step-by-step to avoid errors.

Common Misconceptions

• Simplifying expressions is only necessary for complex expressions.
• Simplifying expressions is only necessary for equations.
• Simplifying expressions is only necessary for certain types of expressions.

Real-World Examples

• Simplifying expressions is essential in physics to solve equations and manipulate mathematical statements.
• Simplifying expressions is critical in engineering to design and analyze systems.
• Simplifying expressions is essential in computer science to optimize algorithms and data structures.

Conclusion

Simplifying expressions is a fundamental skill in algebra that helps us solve equations and manipulate mathematical statements. By understanding the basics of simplifying expressions and avoiding common mistakes, we can simplify even the most complex expressions. Remember to always check your work and use a calculator or computer program if necessary.