Fully Simplify The Expression: -5 X^5 Y^5\left(-6 X Y^3\right ]

Mar 13, 2025 by ADMIN 64 views

$Fully Simplify the Expression: $-5 x^5 y^5\left(-6 x y^3\right)$$

Introduction

Simplifying algebraic expressions is a fundamental concept in mathematics, and it plays a crucial role in solving various mathematical problems. In this article, we will focus on simplifying the given expression $-5 x^5 y^5\left(-6 x y^3\right)$ using the rules of exponents and basic algebraic operations.

Understanding the Expression

The given expression is a product of two terms: $-5 x^5 y^5$ and $-6 x y^3$ . To simplify this expression, we need to apply the rules of exponents and combine like terms.

Applying the Rules of Exponents

When multiplying two terms with the same base, we add their exponents. In this case, we have two terms with the same base $x$ , which are $x^5$ and $x$ . We can add their exponents to get $x^{5+1} = x^6$ . Similarly, we have two terms with the same base $y$ , which are $y^5$ and $y^3$ . We can add their exponents to get $y^{5+3} = y^8$ .

Simplifying the Expression

Now that we have applied the rules of exponents, we can simplify the expression by multiplying the coefficients and combining like terms. The expression becomes:

-5 x^5 y^5\left(-6 x y^3\right) = -5 \cdot -6 \cdot x^5 \cdot x \cdot y^5 \cdot y^3

Combining Like Terms

We can combine like terms by adding their coefficients and multiplying their variables. In this case, we have two terms with the same variable $x$ , which are $x^5$ and $x$ . We can add their coefficients to get $-5 \cdot -6 \cdot x^{5+1} = 30 x^6$ . Similarly, we have two terms with the same variable $y$ , which are $y^5$ and $y^3$ . We can add their coefficients to get $y^{5+3} = y^8$ .

Final Simplification

Now that we have combined like terms, we can simplify the expression further by multiplying the coefficients and combining like terms. The expression becomes:

-5 x^5 y^5\left(-6 x y^3\right) = 30 x^6 y^8

Conclusion

In this article, we have fully simplified the expression $-5 x^5 y^5\left(-6 x y^3\right)$ using the rules of exponents and basic algebraic operations. We have applied the rules of exponents to add the exponents of like bases and combined like terms to simplify the expression. The final simplified expression is $30 x^6 y^8$ .

Tips and Tricks

When simplifying algebraic expressions, it is essential to apply the rules of exponents and combine like terms.
When multiplying two terms with the same base, add their exponents.
When combining like terms, add their coefficients and multiply their variables.
Simplifying algebraic expressions is a crucial skill in mathematics, and it plays a vital role in solving various mathematical problems.

Common Mistakes

Failing to apply the rules of exponents when multiplying two terms with the same base.
Failing to combine like terms when simplifying an expression.
Not following the order of operations when simplifying an expression.

Real-World Applications

Simplifying algebraic expressions is a crucial skill in mathematics, and it has numerous real-world applications in fields such as physics, engineering, and computer science.
Algebraic expressions are used to model real-world problems, and simplifying them is essential to solve these problems.
Simplifying algebraic expressions is a fundamental concept in mathematics, and it plays a vital role in solving various mathematical problems.

Final Thoughts

Simplifying algebraic expressions is a fundamental concept in mathematics, and it plays a crucial role in solving various mathematical problems. In this article, we have fully simplified the expression $-5 x^5 y^5\left(-6 x y^3\right)$ using the rules of exponents and basic algebraic operations. We have applied the rules of exponents to add the exponents of like bases and combined like terms to simplify the expression. The final simplified expression is $30 x^6 y^8$ .

Introduction

Simplifying algebraic expressions is a fundamental concept in mathematics, and it plays a crucial role in solving various mathematical problems. In this article, we will address some of the most frequently asked questions about simplifying algebraic expressions.

Q: What is the order of operations when simplifying algebraic expressions?

A: The order of operations when simplifying algebraic expressions is:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify an expression with multiple variables?

A: To simplify an expression with multiple variables, you need to apply the rules of exponents and combine like terms. For example, if you have the expression $x^2y^2 + 3x^2y^2$ , you can combine like terms by adding the coefficients and multiplying the variables.

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents a value that can change, while a constant is a value that does not change. For example, in the expression $x^2 + 3$ , $x$ is a variable and $3$ is a constant.

Q: How do I simplify an expression with negative exponents?

A: To simplify an expression with negative exponents, you need to apply the rule that $a^{-n} = \frac{1}{a^n}$ . For example, if you have the expression $x^{-2}$ , you can rewrite it as $\frac{1}{x^2}$ .

Q: What is the difference between a monomial and a polynomial?

A: A monomial is an expression with one term, while a polynomial is an expression with two or more terms. For example, in the expression $x^2 + 3$ , $x^2 + 3$ is a polynomial, while $x^2$ is a monomial.

Q: How do I simplify an expression with fractions?

A: To simplify an expression with fractions, you need to apply the rules of fractions and combine like terms. For example, if you have the expression $\frac{x^2}{2} + \frac{3x^2}{2}$ , you can combine like terms by adding the numerators and keeping the denominator the same.

Q: What is the difference between a linear expression and a quadratic expression?

A: A linear expression is an expression with one variable and a degree of one, while a quadratic expression is an expression with one variable and a degree of two. For example, in the expression $x + 3$ , $x + 3$ is a linear expression, while $x^2 + 3$ is a quadratic expression.

Q: How do I simplify an expression with absolute value?

A: To simplify an expression with absolute value, you need to apply the rule that $|a| = a$ if $a \geq 0$ and $|a| = -a$ if $a < 0$ . For example, if you have the expression $|x^2 + 3|$ , you can rewrite it as $x^2 + 3$ if $x^2 + 3 \geq 0$ and $-(x^2 + 3)$ if $x^2 + 3 < 0$ .

Q: What is the difference between a rational expression and an irrational expression?

A: A rational expression is an expression that can be written as a fraction of two polynomials, while an irrational expression is an expression that cannot be written as a fraction of two polynomials. For example, in the expression $\frac{x^2}{2}$ , $\frac{x^2}{2}$ is a rational expression, while $\sqrt{x^2}$ is an irrational expression.

Conclusion

Simplifying algebraic expressions is a fundamental concept in mathematics, and it plays a crucial role in solving various mathematical problems. In this article, we have addressed some of the most frequently asked questions about simplifying algebraic expressions. We hope that this article has provided you with a better understanding of how to simplify algebraic expressions and has helped you to become more confident in your ability to solve mathematical problems.