Rewrite $\frac{1}{(\sqrt{5 X})^3}$ In Exponential Form.A. $(5 X)^{-\frac{3}{2}}$B. $ 5 X − 2 3 5 X^{-\frac{2}{3}} 5 X − 3 2 ​ [/tex]C. $(5 X)^{\frac{2}{8}}$D. $5 X^{\frac{3}{2}}$

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Understanding the Problem

The given expression is $\frac{1}{(\sqrt{5 x})^3}$. We are asked to rewrite this expression in exponential form. To do this, we need to apply the rules of exponents and simplify the expression.

Step 1: Simplify the Expression

The first step is to simplify the expression inside the parentheses. We can start by rewriting the square root as an exponent:

5x=(5x)12\sqrt{5 x} = (5 x)^{\frac{1}{2}}

Now, we can substitute this expression back into the original expression:

1(5x)3=1((5x)12)3\frac{1}{(\sqrt{5 x})^3} = \frac{1}{((5 x)^{\frac{1}{2}})^3}

Step 2: Apply the Power Rule

The next step is to apply the power rule, which states that:

(am)n=am⋅n(a^m)^n = a^{m \cdot n}

In this case, we have:

((5x)12)3=(5x)12⋅3=(5x)32((5 x)^{\frac{1}{2}})^3 = (5 x)^{\frac{1}{2} \cdot 3} = (5 x)^{\frac{3}{2}}

Now, we can substitute this expression back into the original expression:

1(5x)3=1(5x)32\frac{1}{(\sqrt{5 x})^3} = \frac{1}{(5 x)^{\frac{3}{2}}}

Step 3: Rewrite in Exponential Form

The final step is to rewrite the expression in exponential form. We can do this by using the rule:

1am=a−m\frac{1}{a^m} = a^{-m}

In this case, we have:

1(5x)32=(5x)−32\frac{1}{(5 x)^{\frac{3}{2}}} = (5 x)^{-\frac{3}{2}}

Conclusion

Therefore, the correct answer is:

A. $(5 x)^{-\frac{3}{2}}$

This is the exponential form of the given expression.

Comparison with Other Options

Let's compare our answer with the other options:

B. 5x−235 x^{-\frac{2}{3}}

This option is incorrect because the exponent is not correct.

C. $(5 x)^{\frac{2}{8}}$

This option is incorrect because the exponent is not correct.

D. $5 x^{\frac{3}{2}}$

This option is incorrect because the exponent is not correct.

Final Answer

The final answer is:

Q: What is the difference between exponents and exponential form?

A: Exponents are a shorthand way of writing repeated multiplication, while exponential form is a way of writing expressions that involve exponents.

Q: How do I rewrite an expression in exponential form?

A: To rewrite an expression in exponential form, you need to apply the rules of exponents and simplify the expression. This may involve using the power rule, the product rule, and the quotient rule.

Q: What is the power rule for exponents?

A: The power rule for exponents states that:

(am)n=am⋅n(a^m)^n = a^{m \cdot n}

This means that when you raise a power to a power, you multiply the exponents.

Q: What is the product rule for exponents?

A: The product rule for exponents states that:

am⋅an=am+na^m \cdot a^n = a^{m + n}

This means that when you multiply two powers with the same base, you add the exponents.

Q: What is the quotient rule for exponents?

A: The quotient rule for exponents states that:

aman=am−n\frac{a^m}{a^n} = a^{m - n}

This means that when you divide two powers with the same base, you subtract the exponents.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you need to apply the rules of exponents and combine like terms. This may involve using the power rule, the product rule, and the quotient rule.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent indicates that the base is being raised to a power, while a negative exponent indicates that the base is being raised to a power and then taking the reciprocal.

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

A: To evaluate an expression with a negative exponent, you need to take the reciprocal of the base and change the sign of the exponent.

Q: What is the relationship between exponents and logarithms?

A: Exponents and logarithms are inverse operations. This means that if you have an expression with an exponent, you can take the logarithm of the base to get the exponent.

Q: How do I use exponents and logarithms in real-world applications?

A: Exponents and logarithms are used in a wide range of real-world applications, including finance, science, and engineering. They are used to model growth and decay, to calculate interest rates, and to solve problems involving exponential change.

Q: What are some common mistakes to avoid when working with exponents?

A: Some common mistakes to avoid when working with exponents include:

  • Forgetting to apply the power rule
  • Forgetting to apply the product rule
  • Forgetting to apply the quotient rule
  • Not simplifying expressions with exponents
  • Not evaluating expressions with negative exponents correctly

Conclusion

Exponents and exponential form are important concepts in mathematics that have many real-world applications. By understanding the rules of exponents and how to simplify expressions with exponents, you can solve a wide range of problems involving exponential change.