Find The Product. Simplify Your Answer.$ \left(-3z^4\right)\left(-5z^3\right) $

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Understanding the Problem

When dealing with algebraic expressions, simplifying products is a crucial step in solving equations and manipulating expressions. In this case, we are given the expression $\left(-3z^4\right)\left(-5z^3\right)$ and we need to find the product. To simplify this expression, we will use the rules of exponents and the distributive property.

The Distributive Property

The distributive property states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$a(b + c) = ab + ac$

This property can be extended to include exponents by considering the expression $a^m \cdot a^n$ as $a^{m+n}$. We will use this property to simplify the given expression.

Simplifying the Expression

To simplify the expression $\left(-3z^4\right)\left(-5z^3\right)$, we will use the distributive property. We can start by multiplying the coefficients and the variables separately.

Multiplying the Coefficients

The coefficients of the two expressions are $-3$ and $-5$. When we multiply these coefficients, we get:

$-3 \cdot -5 = 15$

Multiplying the Variables

The variables in the two expressions are $z^4$ and $z^3$. When we multiply these variables, we need to add the exponents. This is based on the rule that $a^m \cdot a^n = a^{m+n}$.

$z^4 \cdot z^3 = z^{4+3} = z^7$

Combining the Results

Now that we have multiplied the coefficients and the variables, we can combine the results to get the simplified expression.

$\left(-3z^4\right)\left(-5z^3\right) = 15z^7$

Conclusion

In this article, we have simplified the expression $\left(-3z^4\right)\left(-5z^3\right)$ using the distributive property and the rules of exponents. We have shown that the product of the two expressions is $15z^7$. This result can be used to solve equations and manipulate expressions in algebra.

Tips and Tricks

• When simplifying expressions, it is essential to use the distributive property and the rules of exponents.
• Make sure to multiply the coefficients and the variables separately.
• When adding exponents, remember that $a^m \cdot a^n = a^{m+n}$.

Example Problems

• Simplify the expression $\left(2x^3\right)\left(4x^2\right)$.
• Simplify the expression $\left(-6y^4\right)\left(-2y^3\right)$.

Solutions to Example Problems

• $\left(2x^3\right)\left(4x^2\right) = 8x^{3+2} = 8x^5$
• $\left(-6y^4\right)\left(-2y^3\right) = 12y^{4+3} = 12y^7$

Final Thoughts

Simplifying expressions is a crucial step in solving equations and manipulating expressions in algebra. By using the distributive property and the rules of exponents, we can simplify complex expressions and find the product. Remember to multiply the coefficients and the variables separately and to add exponents when necessary. With practice and patience, you will become proficient in simplifying expressions and solving equations.

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Understanding the Problem

In our previous article, we simplified the expression $\left(-3z^4\right)\left(-5z^3\right)$ using the distributive property and the rules of exponents. We found that the product of the two expressions is $15z^7$. In this article, we will answer some frequently asked questions related to simplifying expressions and provide additional examples.

Q&A

Q: What is the distributive property?

A: The distributive property is a rule in algebra that states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$a(b + c) = ab + ac$

This property can be extended to include exponents by considering the expression $a^m \cdot a^n$ as $a^{m+n}$.

Q: How do I simplify expressions with exponents?

A: To simplify expressions with exponents, you need to multiply the coefficients and the variables separately. When multiplying variables, you need to add the exponents. For example, $z^4 \cdot z^3 = z^{4+3} = z^7$.

Q: What is the rule for multiplying variables with exponents?

A: The rule for multiplying variables with exponents is that $a^m \cdot a^n = a^{m+n}$. This means that when you multiply two variables with the same base, you add the exponents.

Q: Can I simplify expressions with negative coefficients?

A: Yes, you can simplify expressions with negative coefficients. When multiplying two expressions with negative coefficients, the result will have a positive coefficient. For example, $\left(-3z^4\right)\left(-5z^3\right) = 15z^7$.

Q: How do I simplify expressions with multiple variables?

A: To simplify expressions with multiple variables, you need to multiply the coefficients and the variables separately. When multiplying variables, you need to add the exponents. For example, $\left(2x^3y^2\right)\left(4x^2y^3\right) = 8x^{3+2}y^{2+3} = 8x^5y^5$.

Additional Examples

• Simplify the expression $\left(3x^2y^3\right)\left(2x^3y^2\right)$.
• Simplify the expression $\left(-4y^4\right)\left(-3y^3\right)$.

Solutions to Additional Examples

• $\left(3x^2y^3\right)\left(2x^3y^2\right) = 6x^{2+3}y^{3+2} = 6x^5y^5$
• $\left(-4y^4\right)\left(-3y^3\right) = 12y^{4+3} = 12y^7$

Tips and Tricks

• When simplifying expressions, it is essential to use the distributive property and the rules of exponents.
• Make sure to multiply the coefficients and the variables separately.
• When adding exponents, remember that $a^m \cdot a^n = a^{m+n}$.
• Simplify expressions with negative coefficients by multiplying the coefficients and the variables separately.

Conclusion

Simplifying expressions is a crucial step in solving equations and manipulating expressions in algebra. By using the distributive property and the rules of exponents, we can simplify complex expressions and find the product. Remember to multiply the coefficients and the variables separately and to add exponents when necessary. With practice and patience, you will become proficient in simplifying expressions and solving equations.