Simplify The Expression: ${ (x Y^2 Z^3) (x^3 Y^2 Z) X^{[3]} Y Z }$

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. It involves combining like terms, removing unnecessary parentheses, and expressing the expression in its simplest form. In this article, we will simplify the given expression: $(x y^2 z^3) (x^3 y^2 z) x^{[3]} y z$. We will use the properties of exponents, multiplication, and combination of like terms to simplify the expression.

Understanding the Expression

The given expression is a product of three terms: $(x y^2 z^3)$, $(x^3 y^2 z)$, and $x^{[3]} y z$. To simplify the expression, we need to understand the properties of exponents and how they interact with each other. The exponentiation operation is defined as follows:

• $a^m \cdot a^n = a^{m+n}$
• $(a^m)^n = a^{m \cdot n}$
• $a^m \cdot b^n = (a \cdot b)^{m+n}$

Simplifying the Expression

To simplify the expression, we will start by multiplying the first two terms: $(x y^2 z^3)$ and $(x^3 y^2 z)$. We will use the properties of exponents to combine like terms.

import sympy as sp
x, y, z = sp.symbols('x y z')

expr = (x * y2 * z3) * (x3 * y2 * z)

simplified_expr = sp.simplify(expr)

Combining Like Terms

After multiplying the first two terms, we get:

$x^{1+3} y^{2+2} z^{3+1} = x^4 y^4 z^4$

Now, we need to multiply this result with the third term: $x^{[3]} y z$. We will use the properties of exponents to combine like terms.

# Define the third term
third_term = x**[3] * y * z

final_expr = simplified_expr * third_term

Final Simplification

After multiplying the simplified expression with the third term, we get:

$x^{4+3} y^{4+1} z^{4+1} = x^7 y^5 z^5$

Therefore, the simplified expression is $x^7 y^5 z^5$.

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics. In this article, we simplified the given expression: $(x y^2 z^3) (x^3 y^2 z) x^{[3]} y z$. We used the properties of exponents, multiplication, and combination of like terms to simplify the expression. The final simplified expression is $x^7 y^5 z^5$.

Frequently Asked Questions

• What is the property of exponents that states $a^m \cdot a^n = a^{m+n}$?
  • This property states that when we multiply two numbers with the same base, we add their exponents.
• How do we simplify an expression with multiple terms?
  • We start by combining like terms, and then we use the properties of exponents to simplify the expression further.
• What is the final simplified expression?
  • The final simplified expression is $x^7 y^5 z^5$.

Step-by-Step Solution

1. Multiply the first two terms: $(x y^2 z^3)$ and $(x^3 y^2 z)$.
2. Use the properties of exponents to combine like terms.
3. Multiply the simplified expression with the third term: $x^{[3]} y z$.
4. Use the properties of exponents to combine like terms.
5. The final simplified expression is $x^7 y^5 z^5$.

Example Use Cases

• Simplifying algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus.
• The properties of exponents are essential in simplifying algebraic expressions.
• The final simplified expression can be used to solve equations and inequalities.

Code Implementation

import sympy as sp

x, y, z = sp.symbols('x y z')

expr = (x * y2 * z3) * (x3 * y2 * z)

simplified_expr = sp.simplify(expr)

third_term = x**[3] * y * z

final_expr = simplified_expr * third_term

print(final_expr)

Advice

• When simplifying algebraic expressions, use the properties of exponents to combine like terms.
• Start by combining like terms, and then use the properties of exponents to simplify the expression further.
• The final simplified expression can be used to solve equations and inequalities.
  

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Introduction

In our previous article, we simplified the given expression: $(x y^2 z^3) (x^3 y^2 z) x^{[3]} y z$. We used the properties of exponents, multiplication, and combination of like terms to simplify the expression. In this article, we will answer some frequently asked questions related to simplifying algebraic expressions.

Q&A

Q1: What is the property of exponents that states $a^m \cdot a^n = a^{m+n}$?

A1: This property states that when we multiply two numbers with the same base, we add their exponents.

Q2: How do we simplify an expression with multiple terms?

A2: We start by combining like terms, and then we use the properties of exponents to simplify the expression further.

Q3: What is the final simplified expression?

A3: The final simplified expression is $x^7 y^5 z^5$.

Q4: How do we handle negative exponents?

A4: When we have a negative exponent, we can rewrite it as a positive exponent by taking the reciprocal of the base. For example, $a^{-m} = \frac{1}{a^m}$.

Q5: Can we simplify expressions with fractional exponents?

A5: Yes, we can simplify expressions with fractional exponents by using the properties of exponents. For example, $a^{\frac{m}{n}} = \sqrt[n]{a^m}$.

Q6: How do we simplify expressions with radicals?

A6: We can simplify expressions with radicals by using the properties of radicals. For example, $\sqrt{a^m} = a^{\frac{m}{2}}$.

Q7: Can we simplify expressions with absolute values?

A7: Yes, we can simplify expressions with absolute values by using the properties of absolute values. For example, $|a^m| = |a|^m$.

Q8: How do we simplify expressions with complex numbers?

A8: We can simplify expressions with complex numbers by using the properties of complex numbers. For example, $(a+bi)^m = a^m + b^m i^m$.

Step-by-Step Solution

1. Multiply the first two terms: $(x y^2 z^3)$ and $(x^3 y^2 z)$.
2. Use the properties of exponents to combine like terms.
3. Multiply the simplified expression with the third term: $x^{[3]} y z$.
4. Use the properties of exponents to combine like terms.
5. The final simplified expression is $x^7 y^5 z^5$.

Example Use Cases

• Simplifying algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus.
• The properties of exponents are essential in simplifying algebraic expressions.
• The final simplified expression can be used to solve equations and inequalities.

Code Implementation

import sympy as sp

x, y, z = sp.symbols('x y z')

expr = (x * y2 * z3) * (x3 * y2 * z)

simplified_expr = sp.simplify(expr)

third_term = x**[3] * y * z

final_expr = simplified_expr * third_term

print(final_expr)

Advice

• When simplifying algebraic expressions, use the properties of exponents to combine like terms.
• Start by combining like terms, and then use the properties of exponents to simplify the expression further.
• The final simplified expression can be used to solve equations and inequalities.

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics. In this article, we answered some frequently asked questions related to simplifying algebraic expressions. We used the properties of exponents, multiplication, and combination of like terms to simplify the expression. The final simplified expression is $x^7 y^5 z^5$.