Simplify The Expression: $\[ \left(x^5\right)^6 \cdot (-9x^4) \\]

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it is essential to understand the rules and properties that govern these operations. In this article, we will focus on simplifying the given expression $\left(x^5\right)^6 \cdot (-9x^4)$ using the properties of exponents. We will break down the expression step by step, applying the rules of exponentiation to simplify it.

Understanding Exponents

Before we dive into simplifying the expression, let's review the basics of exponents. An exponent is a small number that is written above and to the right of a larger number, indicating how many times the base number should be multiplied by itself. For example, in the expression $x^5$, the exponent 5 indicates that the base number $x$ should be multiplied by itself 5 times: $x \cdot x \cdot x \cdot x \cdot x$.

Simplifying the Expression

Now that we have a basic understanding of exponents, let's apply the rules of exponentiation to simplify the given expression. The expression is $\left(x^5\right)^6 \cdot (-9x^4)$. To simplify this expression, we need to apply the rule of exponentiation that states when we raise a power to another power, we multiply the exponents.

Applying the Rule of Exponentiation

Using the rule of exponentiation, we can rewrite the expression as follows:

$$\left(x^5\right)^6 \cdot (-9x^4) = x^{5 \cdot 6} \cdot (-9x^4)$$

Now, we can simplify the exponent by multiplying the two numbers together:

$$x^{5 \cdot 6} = x^{30}$$

So, the expression becomes:

$$x^{30} \cdot (-9x^4)$$

Combining Like Terms

Now that we have simplified the exponent, we can combine like terms by adding the exponents of the two terms:

$$x^{30} \cdot (-9x^4) = -9x^{30+4}$$

Using the rule of exponentiation that states when we add exponents with the same base, we can simplify the expression further:

$$-9x^{30+4} = -9x^{34}$$

Therefore, the simplified expression is:

$$-9x^{34}$$

Conclusion

In this article, we simplified the expression $\left(x^5\right)^6 \cdot (-9x^4)$ using the properties of exponents. We applied the rule of exponentiation that states when we raise a power to another power, we multiply the exponents, and then combined like terms by adding the exponents of the two terms. The simplified expression is $-9x^{34}$.

Frequently Asked Questions

• What is the rule of exponentiation? The rule of exponentiation states that when we raise a power to another power, we multiply the exponents.
• How do we simplify an expression with exponents? To simplify an expression with exponents, we need to apply the rule of exponentiation and combine like terms by adding the exponents of the two terms.
• What is the simplified expression of $\left(x^5\right)^6 \cdot (-9x^4)$? The simplified expression is $-9x^{34}$.

Final Thoughts

Simplifying algebraic expressions is a crucial skill in mathematics, and it is essential to understand the rules and properties that govern these operations. In this article, we applied the rule of exponentiation and combined like terms to simplify the expression $\left(x^5\right)^6 \cdot (-9x^4)$. We hope that this article has provided you with a better understanding of how to simplify expressions with exponents.

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Introduction

In our previous article, we simplified the expression $\left(x^5\right)^6 \cdot (-9x^4)$ using the properties of exponents. We applied the rule of exponentiation that states when we raise a power to another power, we multiply the exponents, and then combined like terms by adding the exponents of the two terms. In this article, we will answer some frequently asked questions related to simplifying expressions with exponents.

Q&A

Q: What is the rule of exponentiation?

A: The rule of exponentiation states that when we raise a power to another power, we multiply the exponents. For example, in the expression $\left(x^5\right)^6$, the exponent 6 is raised to the power of $x^5$, so we multiply the exponents: $5 \cdot 6 = 30$.

Q: How do we simplify an expression with exponents?

A: To simplify an expression with exponents, we need to apply the rule of exponentiation and combine like terms by adding the exponents of the two terms. For example, in the expression $\left(x^5\right)^6 \cdot (-9x^4)$, we first apply the rule of exponentiation to get $x^{30} \cdot (-9x^4)$, and then combine like terms by adding the exponents: $x^{30+4} = x^{34}$.

Q: What is the difference between a power and an exponent?

A: A power is the base number that is being raised to a certain power, while an exponent is the number that is being raised to a certain power. For example, in the expression $x^5$, $x$ is the base number (the power) and 5 is the exponent.

Q: Can we simplify an expression with negative exponents?

A: Yes, we can simplify an expression with negative exponents by applying the rule of exponentiation that states when we have a negative exponent, we can rewrite it as a positive exponent by flipping the base number. For example, in the expression $x^{-5}$, we can rewrite it as $\frac{1}{x^5}$.

Q: How do we simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, we need to apply the rule of exponentiation and combine like terms by adding the exponents of the two terms. For example, in the expression $x^5 \cdot x^6$, we first apply the rule of exponentiation to get $x^{5+6} = x^{11}$.

Q: Can we simplify an expression with fractional exponents?

A: Yes, we can simplify an expression with fractional exponents by applying the rule of exponentiation that states when we have a fractional exponent, we can rewrite it as a product of two numbers. For example, in the expression $x^{\frac{1}{2}}$, we can rewrite it as $\sqrt{x}$.

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions with exponents. We hope that this article has provided you with a better understanding of how to simplify expressions with exponents and has helped you to overcome any difficulties you may have encountered.

Final Thoughts

Simplifying algebraic expressions is a crucial skill in mathematics, and it is essential to understand the rules and properties that govern these operations. In this article, we applied the rule of exponentiation and combined like terms to simplify expressions with exponents. We hope that this article has provided you with a better understanding of how to simplify expressions with exponents and has helped you to become more confident in your ability to solve algebraic problems.

Additional Resources

• Simplifying Algebraic Expressions
• Exponents and Powers
• Simplifying Expressions with Exponents

Frequently Asked Questions

• What is the rule of exponentiation?
• How do we simplify an expression with exponents?
• What is the difference between a power and an exponent?
• Can we simplify an expression with negative exponents?
• How do we simplify an expression with multiple exponents?
• Can we simplify an expression with fractional exponents?

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