What Is The Base Of The Exponent In The Function $f(x) = 3(\sqrt[3]{8})^{2x}$ When The Function Is Written Using Only Rational Numbers And Is In Simplest Form?$\square$

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What is the Base of the Exponent in the Function f(x)=3(83)2xf(x) = 3(\sqrt[3]{8})^{2x}?

Understanding the Function

The given function is f(x)=3(83)2xf(x) = 3(\sqrt[3]{8})^{2x}. To determine the base of the exponent, we need to simplify the expression and rewrite it using only rational numbers in its simplest form.

Simplifying the Expression

The first step is to simplify the expression inside the parentheses. We know that 83\sqrt[3]{8} can be rewritten as 22, since 23=82^3 = 8. Therefore, we can rewrite the expression as:

f(x)=3(2)2xf(x) = 3(2)^{2x}

Rewriting the Expression Using Rational Numbers

Now, we need to rewrite the expression using only rational numbers. We can do this by expressing 22 as a rational number. Since 2=422 = \frac{4}{2}, we can rewrite the expression as:

f(x)=3(42)2xf(x) = 3(\frac{4}{2})^{2x}

Simplifying the Rational Expression

To simplify the rational expression, we can use the property of exponents that states (ab)n=anbn(\frac{a}{b})^n = \frac{a^n}{b^n}. Applying this property, we get:

f(x)=3(42x22x)f(x) = 3(\frac{4^{2x}}{2^{2x}})

Rewriting the Expression in Simplest Form

Now, we can rewrite the expression in its simplest form by combining the constants:

f(x)=3⋅42x22xf(x) = \frac{3 \cdot 4^{2x}}{2^{2x}}

Determining the Base of the Exponent

The base of the exponent is the number that is raised to the power of 2x2x. In this case, the base is 44. Therefore, the base of the exponent in the function f(x)=3(83)2xf(x) = 3(\sqrt[3]{8})^{2x} is 4\boxed{4}.

Conclusion

In this article, we simplified the given function f(x)=3(83)2xf(x) = 3(\sqrt[3]{8})^{2x} and rewrote it using only rational numbers in its simplest form. We determined that the base of the exponent is 44. This demonstrates the importance of simplifying expressions and rewriting them in their simplest form to understand the underlying mathematical concepts.

Key Takeaways

  • Simplifying expressions is crucial in understanding mathematical concepts.
  • Rewriting expressions using rational numbers can help in identifying the base of the exponent.
  • The base of the exponent is the number that is raised to a power in an exponential expression.

Further Reading

For more information on simplifying expressions and rewriting them using rational numbers, refer to the following resources:

References

Frequently Asked Questions

In this article, we will address some of the most common questions related to the base of the exponent in the function f(x)=3(83)2xf(x) = 3(\sqrt[3]{8})^{2x}.

Q: What is the base of the exponent in the function f(x)=3(83)2xf(x) = 3(\sqrt[3]{8})^{2x}?

A: The base of the exponent in the function f(x)=3(83)2xf(x) = 3(\sqrt[3]{8})^{2x} is 44. This is because 83\sqrt[3]{8} can be rewritten as 22, and 22 can be expressed as a rational number 42\frac{4}{2}.

Q: Why is it important to simplify the expression inside the parentheses?

A: Simplifying the expression inside the parentheses is crucial in understanding the base of the exponent. By rewriting 83\sqrt[3]{8} as 22, we can easily identify the base of the exponent as 44.

Q: Can the base of the exponent be a rational number?

A: Yes, the base of the exponent can be a rational number. In the function f(x)=3(83)2xf(x) = 3(\sqrt[3]{8})^{2x}, the base of the exponent is 44, which is a rational number.

Q: How do I determine the base of the exponent in a given function?

A: To determine the base of the exponent in a given function, you need to simplify the expression inside the parentheses and rewrite it using rational numbers. Then, identify the number that is raised to a power as the base of the exponent.

Q: What is the significance of the base of the exponent in a function?

A: The base of the exponent in a function is crucial in understanding the behavior of the function. It determines the rate at which the function grows or decays.

Q: Can the base of the exponent be a negative number?

A: Yes, the base of the exponent can be a negative number. However, in the function f(x)=3(83)2xf(x) = 3(\sqrt[3]{8})^{2x}, the base of the exponent is a positive number 44.

Q: How do I handle negative bases in exponents?

A: When dealing with negative bases in exponents, you need to consider the properties of exponents. For example, if the base is negative and the exponent is even, the result is positive. If the exponent is odd, the result is negative.

Q: What are some common mistakes to avoid when determining the base of the exponent?

A: Some common mistakes to avoid when determining the base of the exponent include:

  • Not simplifying the expression inside the parentheses
  • Not rewriting the expression using rational numbers
  • Not identifying the number that is raised to a power as the base of the exponent

Conclusion

In this article, we addressed some of the most common questions related to the base of the exponent in the function f(x)=3(83)2xf(x) = 3(\sqrt[3]{8})^{2x}. We hope that this article has provided you with a better understanding of the base of the exponent and how to determine it in a given function.

Key Takeaways

  • Simplifying the expression inside the parentheses is crucial in understanding the base of the exponent.
  • The base of the exponent can be a rational number.
  • The base of the exponent determines the rate at which the function grows or decays.
  • Negative bases in exponents require special consideration.

Further Reading

For more information on the base of the exponent and how to determine it in a given function, refer to the following resources:

References