Which Is An Equivalent Expression For 6 D − 4 21 D 7 \frac{6 D^{-4}}{21 D^7} 21 D 7 6 D − 4 ​ ?A. 2 7 D 11 \frac{2}{7 D^{11}} 7 D 11 2 ​ B. 126 D 11 \frac{126}{d^{11}} D 11 126 ​ C. 0.3 D 11 0.3 D^{11} 0.3 D 11 D. − 2 7 D 3 -\frac{2}{7 D^3} − 7 D 3 2 ​

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Introduction

When dealing with exponential expressions, it's essential to understand the rules of simplification to arrive at equivalent expressions. In this article, we'll focus on simplifying the given expression 6d−421d7\frac{6 d^{-4}}{21 d^7} and explore the different options provided.

Understanding Exponential Expressions

Exponential expressions are a fundamental concept in mathematics, and they follow specific rules for simplification. To simplify an exponential expression, we need to understand the properties of exponents, including the product rule, quotient rule, and power rule.

Product Rule

The product rule states that when multiplying two exponential expressions with the same base, we add the exponents. For example, am⋅an=am+na^m \cdot a^n = a^{m+n}.

Quotient Rule

The quotient rule states that when dividing two exponential expressions with the same base, we subtract the exponents. For example, aman=am−n\frac{a^m}{a^n} = a^{m-n}.

Power Rule

The power rule states that when raising an exponential expression to a power, we multiply the exponents. For example, (am)n=am⋅n(a^m)^n = a^{m \cdot n}.

Simplifying the Given Expression

Now that we've covered the rules of simplification, let's apply them to the given expression 6d−421d7\frac{6 d^{-4}}{21 d^7}.

Step 1: Factor Out the Coefficient

The first step is to factor out the coefficient 6 from the numerator. We can rewrite the expression as 6⋅d−421d7\frac{6 \cdot d^{-4}}{21 d^7}.

Step 2: Simplify the Coefficient

Next, we simplify the coefficient 6 by dividing it by 21. This gives us 27⋅d−4\frac{2}{7} \cdot d^{-4}.

Step 3: Apply the Quotient Rule

Now, we apply the quotient rule to simplify the expression. We subtract the exponents, resulting in 27⋅d−4−7\frac{2}{7} \cdot d^{-4-7}.

Step 4: Simplify the Exponent

Simplifying the exponent, we get 27⋅d−11\frac{2}{7} \cdot d^{-11}.

Step 5: Rewrite the Expression

Finally, we rewrite the expression in a more conventional form, resulting in 27d11\frac{2}{7 d^{11}}.

Comparing the Options

Now that we've simplified the given expression, let's compare it with the options provided.

Option A

Option A is 27d11\frac{2}{7 d^{11}}. This is equivalent to our simplified expression, so it's a valid option.

Option B

Option B is 126d11\frac{126}{d^{11}}. This is not equivalent to our simplified expression, as the coefficient is different.

Option C

Option C is 0.3d110.3 d^{11}. This is not equivalent to our simplified expression, as the coefficient is different and the expression is not in the correct form.

Option D

Option D is −27d3-\frac{2}{7 d^3}. This is not equivalent to our simplified expression, as the exponent is different.

Conclusion

In conclusion, the equivalent expression for 6d−421d7\frac{6 d^{-4}}{21 d^7} is 27d11\frac{2}{7 d^{11}}. This is the correct option, and it's essential to understand the rules of simplification to arrive at this expression.

Frequently Asked Questions

Q: What is the product rule for exponential expressions?

A: The product rule states that when multiplying two exponential expressions with the same base, we add the exponents.

Q: What is the quotient rule for exponential expressions?

A: The quotient rule states that when dividing two exponential expressions with the same base, we subtract the exponents.

Q: What is the power rule for exponential expressions?

A: The power rule states that when raising an exponential expression to a power, we multiply the exponents.

Q: How do I simplify an exponential expression?

A: To simplify an exponential expression, you need to understand the properties of exponents, including the product rule, quotient rule, and power rule.

Final Thoughts

Simplifying exponential expressions is a crucial skill in mathematics, and it's essential to understand the rules of simplification to arrive at equivalent expressions. By following the steps outlined in this article, you can simplify any exponential expression and arrive at the correct answer.

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Introduction

Exponential expressions are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving problems in various fields. In this article, we'll provide a comprehensive Q&A section to help you better understand exponential expressions and how to simplify them.

Q&A Section

Q: What is an exponential expression?

A: An exponential expression is a mathematical expression that involves a base raised to a power. For example, 232^3 is an exponential expression.

Q: What are the rules for simplifying exponential expressions?

A: There are three main rules for simplifying exponential expressions:

  1. Product Rule: When multiplying two exponential expressions with the same base, we add the exponents. For example, am⋅an=am+na^m \cdot a^n = a^{m+n}.
  2. Quotient Rule: When dividing two exponential expressions with the same base, we subtract the exponents. For example, aman=am−n\frac{a^m}{a^n} = a^{m-n}.
  3. Power Rule: When raising an exponential expression to a power, we multiply the exponents. For example, (am)n=am⋅n(a^m)^n = a^{m \cdot n}.

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

A: To simplify an exponential expression with a negative exponent, we can rewrite it as a fraction with a positive exponent. For example, a−m=1ama^{-m} = \frac{1}{a^m}.

Q: Can I simplify an exponential expression with a variable exponent?

A: Yes, you can simplify an exponential expression with a variable exponent by applying the product rule, quotient rule, or power rule. For example, am+n=am⋅ana^{m+n} = a^m \cdot a^n.

Q: How do I simplify an exponential expression with a coefficient?

A: To simplify an exponential expression with a coefficient, you can factor out the coefficient and apply the product rule, quotient rule, or power rule. For example, 2am=2⋅am2a^m = 2 \cdot a^m.

Q: What is the difference between an exponential expression and a polynomial expression?

A: An exponential expression is a mathematical expression that involves a base raised to a power, while a polynomial expression is a mathematical expression that involves variables and coefficients raised to various powers.

Q: Can I simplify an exponential expression with a radical?

A: Yes, you can simplify an exponential expression with a radical by applying the product rule, quotient rule, or power rule. For example, am=am/2\sqrt{a^m} = a^{m/2}.

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

A: To simplify an exponential expression with a fraction, you can apply the quotient rule or the product rule. For example, aman=am−n\frac{a^m}{a^n} = a^{m-n}.

Q: Can I simplify an exponential expression with a trigonometric function?

A: Yes, you can simplify an exponential expression with a trigonometric function by applying the product rule, quotient rule, or power rule. For example, sin⁡(am)=sin⁡(a)⋅am\sin(a^m) = \sin(a) \cdot a^m.

Conclusion

In conclusion, simplifying exponential expressions is a crucial skill in mathematics, and understanding the rules of simplification is essential for solving problems in various fields. By following the Q&A section in this article, you can better understand exponential expressions and how to simplify them.

Frequently Asked Questions

Q: What is the difference between an exponential expression and a logarithmic expression?

A: An exponential expression is a mathematical expression that involves a base raised to a power, while a logarithmic expression is a mathematical expression that involves the inverse operation of exponentiation.

Q: Can I simplify an exponential expression with a complex number?

A: Yes, you can simplify an exponential expression with a complex number by applying the product rule, quotient rule, or power rule.

Q: How do I simplify an exponential expression with a matrix?

A: To simplify an exponential expression with a matrix, you can apply the product rule, quotient rule, or power rule.

Final Thoughts

Simplifying exponential expressions is a crucial skill in mathematics, and understanding the rules of simplification is essential for solving problems in various fields. By following the Q&A section in this article, you can better understand exponential expressions and how to simplify them.

Additional Resources