Select The Expression That Is Equivalent To $\left(3 D^7\right)^3$.$\[ \begin{array}{cccc} \text{A. } 9 D^{10} & \text{B. } 27 D^{10} & \text{C. } 27 D^{21} & \text{D. } 3 D^{21} \\ \end{array} \\]

Understanding Exponents and Their Rules

Exponents are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will focus on simplifying exponential expressions, specifically the expression $\left(3 d^7\right)^3$. We will explore the rules of exponents, apply them to the given expression, and determine the equivalent expression.

The Rules of Exponents

Before we dive into simplifying the expression, let's review the rules of exponents:

• Product of Powers Rule: When multiplying two powers with the same base, add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.
• Power of a Power Rule: When raising a power to another power, multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.
• Zero Exponent Rule: Any non-zero number raised to the power of zero is equal to 1. For example, $a^0 = 1$.

Simplifying the Expression

Now that we have reviewed the rules of exponents, let's apply them to the expression $\left(3 d^7\right)^3$. Using the Power of a Power Rule, we can simplify the expression as follows:

$$\left(3 d^7\right)^3 = 3^3 \cdot (d^7)^3 = 27 \cdot d^{7 \cdot 3} = 27 \cdot d^{21}$$

Therefore, the equivalent expression to $\left(3 d^7\right)^3$ is $27 d^{21}$.

Conclusion

In this article, we have simplified the exponential expression $\left(3 d^7\right)^3$ using the rules of exponents. We have applied the Power of a Power Rule to simplify the expression and have determined that the equivalent expression is $27 d^{21}$. This example demonstrates the importance of understanding the rules of exponents and how to apply them to simplify complex expressions.

Answer

The correct answer is C. $27 d^{21}$.

Additional Examples

To further reinforce your understanding of simplifying exponential expressions, let's consider a few more examples:

• Example 1: Simplify the expression $(2x^3)^2$.
  • Using the Power of a Power Rule, we can simplify the expression as follows: $(2x^3)^2 = 2^2 \cdot (x^3)^2 = 4 \cdot x^{3 \cdot 2} = 4 \cdot x^6$
  • Therefore, the equivalent expression to $(2x^3)^2$ is $4x^6$.
• Example 2: Simplify the expression $(3y^4)^3$.
  • Using the Power of a Power Rule, we can simplify the expression as follows: $(3y^4)^3 = 3^3 \cdot (y^4)^3 = 27 \cdot y^{4 \cdot 3} = 27 \cdot y^{12}$
  • Therefore, the equivalent expression to $(3y^4)^3$ is $27y^{12}$.

Practice Problems

To further practice simplifying exponential expressions, try the following problems:

• Simplify the expression $(4z^2)^3$.
• Simplify the expression $(2w^5)^2$.
• Simplify the expression $(3x^4)^4$.

Solutions

• Problem 1: $(4z^2)^3 = 4^3 \cdot (z^2)^3 = 64 \cdot z^{2 \cdot 3} = 64 \cdot z^6$
• Problem 2: $(2w^5)^2 = 2^2 \cdot (w^5)^2 = 4 \cdot w^{5 \cdot 2} = 4 \cdot w^{10}$
• Problem 3: $(3x^4)^4 = 3^4 \cdot (x^4)^4 = 81 \cdot x^{4 \cdot 4} = 81 \cdot x^{16}$
  Frequently Asked Questions: Simplifying Exponential Expressions ====================================================================

Q: What is the rule for simplifying exponential expressions?

A: The rule for simplifying exponential expressions is based on the Power of a Power Rule, which states that when raising a power to another power, multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.

Q: How do I simplify an expression with multiple bases?

A: To simplify an expression with multiple bases, you can use the Product of Powers Rule, which states that when multiplying two powers with the same base, add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.

Q: What is the zero exponent rule?

A: The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1. For example, $a^0 = 1$.

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

A: To simplify an expression with a negative exponent, you can use the rule that states $a^{-n} = \frac{1}{a^n}$. For example, $x^{-3} = \frac{1}{x^3}$.

Q: Can I simplify an expression with a fractional exponent?

A: Yes, you can simplify an expression with a fractional exponent. To do this, you can use the rule that states $(a^m)^n = a^{m \cdot n}$. For example, $(x^{\frac{1}{2}})^3 = x^{\frac{3}{2}}$.

Q: How do I simplify an expression with a variable in the exponent?

A: To simplify an expression with a variable in the exponent, you can use the same rules as before. For example, $(2x^3)^2 = 2^2 \cdot (x^3)^2 = 4 \cdot x^6$.

Q: Can I simplify an expression with a coefficient in the exponent?

A: Yes, you can simplify an expression with a coefficient in the exponent. To do this, you can use the rule that states $a^m \cdot a^n = a^{m+n}$. For example, $(3x^2)^3 = 3^3 \cdot (x^2)^3 = 27 \cdot x^6$.

Q: How do I simplify an expression with multiple variables in the exponent?

A: To simplify an expression with multiple variables in the exponent, you can use the same rules as before. For example, $(2x^3y^2)^3 = 2^3 \cdot (x^3)^3 \cdot (y^2)^3 = 8 \cdot x^9 \cdot y^6$.

Q: Can I simplify an expression with a negative coefficient in the exponent?

A: Yes, you can simplify an expression with a negative coefficient in the exponent. To do this, you can use the rule that states $a^m \cdot a^n = a^{m+n}$. For example, $(-2x^3)^3 = (-2)^3 \cdot (x^3)^3 = -8 \cdot x^9$.

Q: How do I simplify an expression with a variable in the base and a coefficient in the exponent?

A: To simplify an expression with a variable in the base and a coefficient in the exponent, you can use the same rules as before. For example, $(3x^2)^3 = 3^3 \cdot (x^2)^3 = 27 \cdot x^6$.

Q: Can I simplify an expression with a fractional coefficient in the exponent?

A: Yes, you can simplify an expression with a fractional coefficient in the exponent. To do this, you can use the rule that states $a^m \cdot a^n = a^{m+n}$. For example, $(\frac{1}{2}x^3)^2 = (\frac{1}{2})^2 \cdot (x^3)^2 = \frac{1}{4} \cdot x^6$.

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

A: To simplify an expression with a negative variable in the exponent, you can use the rule that states $a^{-n} = \frac{1}{a^n}$. For example, $(-x)^3 = -x^3$.

Q: Can I simplify an expression with a variable in the exponent and a coefficient in the base?

A: Yes, you can simplify an expression with a variable in the exponent and a coefficient in the base. To do this, you can use the same rules as before. For example, $(2x^3)^3 = 2^3 \cdot (x^3)^3 = 8 \cdot x^9$.

Q: How do I simplify an expression with multiple variables in the base and a coefficient in the exponent?

A: To simplify an expression with multiple variables in the base and a coefficient in the exponent, you can use the same rules as before. For example, $(2x^3y^2)^3 = 2^3 \cdot (x^3)^3 \cdot (y^2)^3 = 8 \cdot x^9 \cdot y^6$.

Q: Can I simplify an expression with a variable in the base and a fractional coefficient in the exponent?

A: Yes, you can simplify an expression with a variable in the base and a fractional coefficient in the exponent. To do this, you can use the same rules as before. For example, $(\frac{1}{2}x^3)^2 = (\frac{1}{2})^2 \cdot (x^3)^2 = \frac{1}{4} \cdot x^6$.

Q: How do I simplify an expression with multiple variables in the base and a fractional coefficient in the exponent?

A: To simplify an expression with multiple variables in the base and a fractional coefficient in the exponent, you can use the same rules as before. For example, $(\frac{1}{2}x^3y^2)^2 = (\frac{1}{2})^2 \cdot (x^3)^2 \cdot (y^2)^2 = \frac{1}{4} \cdot x^6 \cdot y^4$.

Conclusion

In this article, we have answered some of the most frequently asked questions about simplifying exponential expressions. We have covered topics such as the Power of a Power Rule, the Product of Powers Rule, and the zero exponent rule. We have also provided examples and practice problems to help you understand and apply these rules.