Choose The Correct Simplification Of The Expression $\left(5 X^4 Y^2 Z^2\right)\left(3 X^4 Y^3 Z^5\right$\].A. $15 X^{16} Y^6 Z^{10}$B. $15 X^8 Y^5 Z^7$C. $8 X^8 Y^5 Z^7$D. $8 X^{16} Y^6 Z^{10}$

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will focus on simplifying the expression $\left(5 x^4 y^2 z^2\right)\left(3 x^4 y^3 z^5\right)$, which is a classic example of a product of two algebraic expressions. We will explore the different options available and choose the correct simplification.

Understanding Exponents

Before we dive into the simplification process, it's essential to understand the concept of exponents. Exponents are a shorthand way of representing repeated multiplication. For example, $x^4$ means $x \times x \times x \times x$, and $y^3$ means $y \times y \times y$. When we multiply two expressions with the same base, we add the exponents. For example, $x^4 \times x^3 = x^{4+3} = x^7$.

Simplifying the Expression

Now that we have a good understanding of exponents, let's simplify the expression $\left(5 x^4 y^2 z^2\right)\left(3 x^4 y^3 z^5\right)$. To simplify this expression, we need to multiply the coefficients (the numbers in front of the variables) and add the exponents of the variables.

Step 1: Multiply the Coefficients

The coefficients of the two expressions are 5 and 3, respectively. To multiply these coefficients, we simply multiply them together: $5 \times 3 = 15$.

Step 2: Add the Exponents

Now, let's add the exponents of the variables. The exponents of $x$ are 4 and 4, respectively. To add these exponents, we simply add them together: $4 + 4 = 8$. The exponents of $y$ are 2 and 3, respectively. To add these exponents, we simply add them together: $2 + 3 = 5$. The exponents of $z$ are 2 and 5, respectively. To add these exponents, we simply add them together: $2 + 5 = 7$.

Step 3: Write the Simplified Expression

Now that we have multiplied the coefficients and added the exponents, we can write the simplified expression: $15 x^8 y^5 z^7$.

Choosing the Correct Simplification

Now that we have simplified the expression, let's compare it to the options available. The correct simplification is:

• A. $15 x^{16} y^6 z^{10}$: This option is incorrect because the exponent of $x$ is 16, not 8.
• B. $15 x^8 y^5 z^7$: This option is correct because it matches the simplified expression we derived.
• C. $8 x^8 y^5 z^7$: This option is incorrect because the coefficient is 8, not 15.
• D. $8 x^{16} y^6 z^{10}$: This option is incorrect because the coefficient is 8, not 15, and the exponent of $x$ is 16, not 8.

Conclusion

In conclusion, the correct simplification of the expression $\left(5 x^4 y^2 z^2\right)\left(3 x^4 y^3 z^5\right)$ is $15 x^8 y^5 z^7$. This simplification was achieved by multiplying the coefficients and adding the exponents of the variables. We compared the simplified expression to the options available and chose the correct simplification.

Final Answer

Introduction

In our previous article, we explored the concept of simplifying algebraic expressions, focusing on the expression $\left(5 x^4 y^2 z^2\right)\left(3 x^4 y^3 z^5\right)$. We derived the correct simplification, which is $15 x^8 y^5 z^7$. In this article, we will provide a Q&A guide to help you better understand the concept of simplifying algebraic expressions.

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to identify the coefficients and the variables. The coefficients are the numbers in front of the variables, and the variables are the letters or symbols that represent the unknown values.

Q: How do I multiply the coefficients?

A: To multiply the coefficients, you simply multiply them together. For example, if you have two coefficients, 5 and 3, you would multiply them together to get 15.

Q: How do I add the exponents?

A: To add the exponents, you simply add the exponents of the variables. For example, if you have two variables, $x^4$ and $x^3$, you would add the exponents to get $x^{4+3} = x^7$.

Q: What is the rule for multiplying variables with the same base?

A: The rule for multiplying variables with the same base is to add the exponents. For example, if you have two variables, $x^4$ and $x^3$, you would add the exponents to get $x^{4+3} = x^7$.

Q: Can I simplify an expression with variables that have different bases?

A: Yes, you can simplify an expression with variables that have different bases. However, you cannot add the exponents of variables with different bases. For example, if you have two variables, $x^4$ and $y^3$, you cannot add the exponents to get $x^{4+3} = x^7$.

Q: How do I simplify an expression with parentheses?

A: To simplify an expression with parentheses, you need to follow the order of operations (PEMDAS). First, evaluate the expressions inside the parentheses, and then simplify the expression.

Q: Can I simplify an expression with negative exponents?

A: Yes, you can simplify an expression with negative exponents. To simplify an expression with a negative exponent, you need to rewrite the expression with a positive exponent. For example, if you have an expression with a negative exponent, $x^{-3}$, you can rewrite it as $\frac{1}{x^3}$.

Q: How do I simplify an expression with fractions?

A: To simplify an expression with fractions, you need to follow the rules of fraction multiplication. To multiply fractions, you multiply the numerators and the denominators separately. For example, if you have two fractions, $\frac{1}{2}$ and $\frac{3}{4}$, you would multiply the numerators and the denominators to get $\frac{1 \times 3}{2 \times 4} = \frac{3}{8}$.

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill for students and professionals alike. By following the rules of exponent multiplication and fraction multiplication, you can simplify complex expressions and arrive at the correct solution. We hope this Q&A guide has helped you better understand the concept of simplifying algebraic expressions.

Final Tips

• Always follow the order of operations (PEMDAS) when simplifying expressions.
• Use parentheses to group expressions and make them easier to simplify.
• Be careful when working with negative exponents and fractions.
• Practice, practice, practice! The more you practice simplifying expressions, the more comfortable you will become with the rules and procedures.