Simplify The Expression: ${ \frac{6^3 \cdot 6 7}{6 4} }$

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. One of the most common techniques used to simplify expressions is exponentiation. In this article, we will focus on simplifying the given expression $\frac{6^3 \cdot 6^7}{6^4}$ using the properties of exponents.

Understanding Exponents

Before we dive into simplifying the expression, let's briefly review the concept of exponents. An exponent is a small number that is placed above and to the right of a base number. It tells us how many times to multiply the base number by itself. For example, $6^3$ means $6$ multiplied by itself $3$ times, which is equal to $6 \cdot 6 \cdot 6 = 216$.

Properties of Exponents

There are several properties of exponents that we will use to simplify the given expression. These properties include:

• Product of Powers: When we multiply two powers with the same base, we add their exponents. For example, $a^m \cdot a^n = a^{m+n}$.
• Power of a Power: When we raise a power to another power, we multiply their exponents. For example, $(a^m)^n = a^{m \cdot n}$.
• Quotient of Powers: When we divide two powers with the same base, we subtract their exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.

Simplifying the Expression

Now that we have reviewed the properties of exponents, let's simplify the given expression $\frac{6^3 \cdot 6^7}{6^4}$.

Using the Product of Powers property, we can simplify the numerator as follows:

$$6^3 \cdot 6^7 = 6^{3+7} = 6^{10}$$

Now, we can rewrite the original expression as:

$$\frac{6^{10}}{6^4}$$

Using the Quotient of Powers property, we can simplify the expression as follows:

$$\frac{6^{10}}{6^4} = 6^{10-4} = 6^6$$

Therefore, the simplified expression is $\boxed{6^6}$.

Conclusion

In this article, we have simplified the expression $\frac{6^3 \cdot 6^7}{6^4}$ using the properties of exponents. We have reviewed the concept of exponents and the properties of exponents, including the product of powers, power of a power, and quotient of powers. We have then applied these properties to simplify the given expression. The simplified expression is $\boxed{6^6}$.

Final Answer

The final answer is $\boxed{6^6}$.

Step-by-Step Solution

Here are the step-by-step solutions to simplify the expression:

1. Simplify the numerator using the Product of Powers property: $6^3 \cdot 6^7 = 6^{3+7} = 6^{10}$
2. Rewrite the original expression: $\frac{6^{10}}{6^4}$
3. Simplify the expression using the Quotient of Powers property: $\frac{6^{10}}{6^4} = 6^{10-4} = 6^6$

Frequently Asked Questions

• What is the simplified expression of $\frac{6^3 \cdot 6^7}{6^4}$?
• How do we simplify expressions using the properties of exponents?
• What are the properties of exponents that we use to simplify expressions?

Answer to Frequently Asked Questions

• The simplified expression is $\boxed{6^6}$.
• We simplify expressions using the properties of exponents, including the product of powers, power of a power, and quotient of powers.
• The properties of exponents that we use to simplify expressions include the product of powers, power of a power, and quotient of powers.
  

Introduction

In our previous article, we simplified the expression $\frac{6^3 \cdot 6^7}{6^4}$ using the properties of exponents. In this article, we will answer some frequently asked questions related to simplifying expressions with exponents.

Q&A

Q: What is the simplified expression of $\frac{6^3 \cdot 6^7}{6^4}$?

A: The simplified expression is $\boxed{6^6}$.

Q: How do we simplify expressions using the properties of exponents?

A: We simplify expressions using the properties of exponents, including the product of powers, power of a power, and quotient of powers. These properties are:

• Product of Powers: When we multiply two powers with the same base, we add their exponents. For example, $a^m \cdot a^n = a^{m+n}$.
• Power of a Power: When we raise a power to another power, we multiply their exponents. For example, $(a^m)^n = a^{m \cdot n}$.
• Quotient of Powers: When we divide two powers with the same base, we subtract their exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.

Q: What are the properties of exponents that we use to simplify expressions?

A: The properties of exponents that we use to simplify expressions include the product of powers, power of a power, and quotient of powers.

Q: How do we apply the product of powers property to simplify expressions?

A: To apply the product of powers property, we add the exponents of the two powers with the same base. For example, $a^m \cdot a^n = a^{m+n}$.

Q: How do we apply the power of a power property to simplify expressions?

A: To apply the power of a power property, we multiply the exponents of the two powers. For example, $(a^m)^n = a^{m \cdot n}$.

Q: How do we apply the quotient of powers property to simplify expressions?

A: To apply the quotient of powers property, we subtract the exponents of the two powers with the same base. For example, $\frac{a^m}{a^n} = a^{m-n}$.

Q: Can we simplify expressions with different bases?

A: Yes, we can simplify expressions with different bases using the properties of exponents. However, we need to make sure that the bases are the same before we can apply the properties.

Q: How do we simplify expressions with negative exponents?

A: To simplify expressions with negative exponents, we can use the property of negative exponents, which states that $a^{-m} = \frac{1}{a^m}$.

Conclusion

In this article, we have answered some frequently asked questions related to simplifying expressions with exponents. We have reviewed the properties of exponents, including the product of powers, power of a power, and quotient of powers. We have also provided examples of how to apply these properties to simplify expressions.

Final Answer

The final answer is $\boxed{6^6}$.

Step-by-Step Solution

Here are the step-by-step solutions to simplify the expression:

1. Simplify the numerator using the Product of Powers property: $6^3 \cdot 6^7 = 6^{3+7} = 6^{10}$
2. Rewrite the original expression: $\frac{6^{10}}{6^4}$
3. Simplify the expression using the Quotient of Powers property: $\frac{6^{10}}{6^4} = 6^{10-4} = 6^6$

Frequently Asked Questions

• What is the simplified expression of $\frac{6^3 \cdot 6^7}{6^4}$?
• How do we simplify expressions using the properties of exponents?
• What are the properties of exponents that we use to simplify expressions?

Answer to Frequently Asked Questions

• The simplified expression is $\boxed{6^6}$.
• We simplify expressions using the properties of exponents, including the product of powers, power of a power, and quotient of powers.
• The properties of exponents that we use to simplify expressions include the product of powers, power of a power, and quotient of powers.