Simplify The Expression:$\left(4a^4\right)\left(-5a^{-6}\right) =$

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 expression $\left(4a^4\right)\left(-5a^{-6}\right)$ using the rules of exponents and algebraic manipulation.

Understanding Exponents

Before we dive into simplifying the expression, let's review the basics of exponents. An exponent is a small number that is placed above and to the right of a number or a variable. It represents the power to which the base is raised. For example, in the expression $a^4$, the exponent 4 represents the power to which the base $a$ is raised.

The Product of Powers Rule

One of the most important rules in algebra is the product of powers rule, which states that when we multiply two powers with the same base, we add their exponents. Mathematically, this can be represented as:

$a^m \cdot a^n = a^{m+n}$

Simplifying the Expression

Now that we have reviewed the basics of exponents and the product of powers rule, let's simplify the expression $\left(4a^4\right)\left(-5a^{-6}\right)$.

To simplify this expression, we will use the product of powers rule. We can rewrite the expression as:

$\left(4a^4\right)\left(-5a^{-6}\right) = 4 \cdot -5 \cdot a^4 \cdot a^{-6}$

Using the product of powers rule, we can add the exponents of the two powers with the same base:

$a^4 \cdot a^{-6} = a^{4+(-6)} = a^{-2}$

Now, we can rewrite the expression as:

$4 \cdot -5 \cdot a^{-2}$

Simplifying the Coefficients

The expression $4 \cdot -5$ can be simplified by multiplying the two numbers:

$4 \cdot -5 = -20$

So, the expression becomes:

$-20 \cdot a^{-2}$

Final Simplification

The expression $-20 \cdot a^{-2}$ can be rewritten as:

$\frac{-20}{a^2}$

This is the final simplified form of the expression.

Conclusion

In this article, we have simplified the expression $\left(4a^4\right)\left(-5a^{-6}\right)$ using the rules of exponents and algebraic manipulation. We have reviewed the basics of exponents and the product of powers rule, and we have applied these rules to simplify the expression. The final simplified form of the expression is $\frac{-20}{a^2}$.

Common Mistakes to Avoid

When simplifying expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not using the product of powers rule: When multiplying two powers with the same base, make sure to add their exponents.
• Not simplifying the coefficients: Make sure to simplify the coefficients by multiplying or dividing them.
• Not rewriting the expression in a simpler form: Make sure to rewrite the expression in a simpler form by using the rules of exponents and algebraic manipulation.

Practice Problems

To practice simplifying expressions, try the following problems:

• Simplify the expression $\left(3a^2\right)\left(-2a^{-3}\right)$.
• Simplify the expression $\left(4b^4\right)\left(-5b^{-2}\right)$.
• Simplify the expression $\left(2c^3\right)\left(-3c^{-1}\right)$.

Conclusion

Q&A: Simplifying Expressions

In this article, we will answer some common questions related to simplifying expressions.

Q: What is the product of powers rule?

A: The product of powers rule states that when we multiply two powers with the same base, we add their exponents. Mathematically, this can be represented as:

$a^m \cdot a^n = a^{m+n}$

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, follow these steps:

1. Identify the base and the exponents in the expression.
2. Use the product of powers rule to add the exponents of the two powers with the same base.
3. Simplify the coefficients by multiplying or dividing them.
4. Rewrite the expression in a simpler form by using the rules of exponents and algebraic manipulation.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent represents a power to which the base is raised, while a negative exponent represents a reciprocal of the base. For example, in the expression $a^4$, the exponent 4 represents the power to which the base $a$ is raised. In the expression $a^{-4}$, the exponent -4 represents a reciprocal of the base $a$.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, follow these steps:

1. Rewrite the expression with a positive exponent by using the rule $a^{-n} = \frac{1}{a^n}$.
2. Simplify the expression by using the product of powers rule and the rules of algebraic manipulation.

Q: What is the rule for dividing powers with the same base?

A: The rule for dividing powers with the same base is:

$\frac{a^m}{a^n} = a^{m-n}$

Q: How do I simplify an expression with a fraction?

A: To simplify an expression with a fraction, follow these steps:

1. Simplify the numerator and denominator separately.
2. Use the rule for dividing powers with the same base to simplify the expression.
3. Rewrite the expression in a simpler form by using the rules of exponents and algebraic manipulation.

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

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

• Not using the product of powers rule
• Not simplifying the coefficients
• Not rewriting the expression in a simpler form
• Not using the rule for dividing powers with the same base

Q: How can I practice simplifying expressions?

A: You can practice simplifying expressions by:

• Working on practice problems
• Using online resources and worksheets
• Asking a teacher or tutor for help
• Joining a study group or online community to practice and learn from others

Conclusion

Simplifying expressions is a crucial skill in algebra that helps us solve equations and manipulate mathematical statements. By understanding the basics of exponents and the product of powers rule, we can simplify expressions and rewrite them in a simpler form. Remember to avoid common mistakes and practice simplifying expressions to become proficient in algebraic manipulation.

Practice Problems

To practice simplifying expressions, try the following problems:

• Simplify the expression $\left(3a^2\right)\left(-2a^{-3}\right)$.
• Simplify the expression $\left(4b^4\right)\left(-5b^{-2}\right)$.
• Simplify the expression $\left(2c^3\right)\left(-3c^{-1}\right)$.

Additional Resources

For more information on simplifying expressions, check out the following resources:

• Khan Academy: Simplifying Expressions
• Mathway: Simplifying Expressions
• IXL: Simplifying Expressions

Conclusion

Simplifying expressions is a crucial skill in algebra that helps us solve equations and manipulate mathematical statements. By understanding the basics of exponents and the product of powers rule, we can simplify expressions and rewrite them in a simpler form. Remember to avoid common mistakes and practice simplifying expressions to become proficient in algebraic manipulation.