What Is The Value Of $n$ In The Simplified Expression? ( 4 K 7 ) 3 = 4 N ⋅ ( K 7 ) 3 = 64 K 21 \left(4 K^7\right)^3 = 4^n \cdot \left(k^7\right)^3 = 64 K^{21} ( 4 K 7 ) 3 = 4 N ⋅ ( K 7 ) 3 = 64 K 21 N = N = N =

Understanding the Problem

When dealing with exponents, it's essential to understand the rules of exponentiation. The given expression involves raising a power to another power, which can be simplified using the power of a power rule. This rule states that when a power is raised to another power, the exponents are multiplied. In this case, we have $\left(4 k^7\right)^3$, which can be simplified using this rule.

Applying the Power of a Power Rule

To simplify the expression $\left(4 k^7\right)^3$, we need to apply the power of a power rule. This rule states that when a power is raised to another power, the exponents are multiplied. In this case, we have:

$\left(4 k^7\right)^3 = 4^3 \cdot \left(k^7\right)^3$

Using the power of a power rule, we can simplify this expression as follows:

$4^3 = 64$

$\left(k^7\right)^3 = k^{21}$

Therefore, the simplified expression is:

$64 k^{21}$

Finding the Value of $n$

Now that we have simplified the expression, we need to find the value of $n$. The given expression is:

$4^n \cdot \left(k^7\right)^3 = 64 k^{21}$

We can see that the simplified expression is $64 k^{21}$. Since $64 = 4^3$, we can rewrite the expression as:

$4^3 \cdot k^{21} = 64 k^{21}$

Comparing this expression with the given expression, we can see that:

$n = 3$

Conclusion

In this problem, we were given an expression involving exponents and asked to find the value of $n$. We simplified the expression using the power of a power rule and compared it with the given expression to find the value of $n$. The final answer is $n = 3$.

Step-by-Step Solution

Here's a step-by-step solution to the problem:

1. Simplify the expression $\left(4 k^7\right)^3$ using the power of a power rule.
2. Rewrite the simplified expression as $4^3 \cdot k^{21}$.
3. Compare the simplified expression with the given expression to find the value of $n$.

Frequently Asked Questions

• What is the power of a power rule? The power of a power rule states that when a power is raised to another power, the exponents are multiplied.
• How do you simplify an expression involving exponents? To simplify an expression involving exponents, you need to apply the power of a power rule and multiply the exponents.
• What is the value of $n$ in the simplified expression? The value of $n$ in the simplified expression is $3$.

Final Answer

The final answer is $\boxed{3}$.

Understanding Exponents and Simplifying Expressions

Exponents and simplifying expressions are fundamental concepts in mathematics that are used to represent and manipulate complex mathematical expressions. In this article, we will answer some frequently asked questions about exponents and simplifying expressions.

Q: What is the power of a power rule?

A: The power of a power rule is a mathematical rule that states that when a power is raised to another power, the exponents are multiplied. This rule is used to simplify expressions involving exponents.

Q: How do you simplify an expression involving exponents?

A: To simplify an expression involving exponents, you need to apply the power of a power rule and multiply the exponents. You also need to apply other rules such as the product of powers rule and the quotient of powers rule.

Q: What is the product of powers rule?

A: The product of powers rule is a mathematical rule that states that when you multiply two or more powers with the same base, you add the exponents. This rule is used to simplify expressions involving the product of powers.

Q: What is the quotient of powers rule?

A: The quotient of powers rule is a mathematical rule that states that when you divide two or more powers with the same base, you subtract the exponents. This rule is used to simplify expressions involving the quotient of powers.

Q: How do you handle negative exponents?

A: Negative exponents can be handled by rewriting them as positive exponents with a reciprocal base. For example, $a^{-n} = \frac{1}{a^n}$.

Q: How do you handle fractional exponents?

A: Fractional exponents can be handled by rewriting them as roots. For example, $a^{\frac{1}{n}} = \sqrt[n]{a}$.

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

A: A power is a mathematical expression that represents a quantity raised to a certain power, while an exponent is the number that is raised to a certain power.

Q: How do you simplify expressions involving radicals?

A: To simplify expressions involving radicals, you need to apply the rules of radicals, such as the product of radicals rule and the quotient of radicals rule.

Q: What is the product of radicals rule?

A: The product of radicals rule is a mathematical rule that states that when you multiply two or more radicals with the same index, you multiply the radicands and keep the index the same.

Q: What is the quotient of radicals rule?

A: The quotient of radicals rule is a mathematical rule that states that when you divide two or more radicals with the same index, you divide the radicands and keep the index the same.

Q: How do you handle complex numbers involving exponents?

A: Complex numbers involving exponents can be handled by applying the rules of exponents and the rules of complex numbers.

Q: What is the difference between a real number and a complex number?

A: A real number is a number that can be expressed as a finite decimal or fraction, while a complex number is a number that can be expressed as a sum of a real number and an imaginary number.

Q: How do you simplify expressions involving complex numbers?

A: To simplify expressions involving complex numbers, you need to apply the rules of complex numbers, such as the product of complex numbers rule and the quotient of complex numbers rule.

Q: What is the product of complex numbers rule?

A: The product of complex numbers rule is a mathematical rule that states that when you multiply two or more complex numbers, you multiply the real parts and the imaginary parts separately.

Q: What is the quotient of complex numbers rule?

A: The quotient of complex numbers rule is a mathematical rule that states that when you divide two or more complex numbers, you divide the real parts and the imaginary parts separately.

Final Answer

The final answer is that exponents and simplifying expressions are fundamental concepts in mathematics that are used to represent and manipulate complex mathematical expressions. By understanding the rules of exponents and simplifying expressions, you can simplify complex expressions and solve mathematical problems with ease.