What Is The Simplified Value Of The Exponential Expression $27^{\frac{1}{3}}$?A. $\frac{1}{3}$ B. \$\frac{1}{9}$[/tex\] C. 3 D. 9

Feb 28, 2025 by ADMIN 140 views

$What is the Simplified Value of the Exponential Expression $27^{\frac{1}{3}}$?$

Understanding Exponential Expressions

Exponential expressions are mathematical expressions that involve the use of exponents, which are shorthand notations for repeated multiplication. In the given expression $27^{\frac{1}{3}}$, the base is 27 and the exponent is $\frac{1}{3}$. To simplify this expression, we need to understand the concept of fractional exponents and how they relate to the base.

Fractional Exponents

Fractional exponents are a way of expressing a number as a power of another number. In the expression $27^{\frac{1}{3}}$, the exponent $\frac{1}{3}$ can be interpreted as a fraction, where the numerator is 1 and the denominator is 3. This means that the expression can be rewritten as $\sqrt[3]{27}$, where the cube root of 27 is being taken.

Simplifying the Expression

To simplify the expression $\sqrt[3]{27}$, we need to find the cube root of 27. The cube root of a number is a value that, when multiplied by itself twice, gives the original number. In this case, the cube root of 27 is 3, because $3 \times 3 \times 3 = 27$.

Conclusion

Therefore, the simplified value of the exponential expression $27^{\frac{1}{3}}$ is 3. This is because the cube root of 27 is 3, and the expression can be rewritten as $\sqrt[3]{27}$.

Real-World Applications

Exponential expressions like $27^{\frac{1}{3}}$ have many real-world applications in fields such as science, engineering, and finance. For example, in physics, the exponential decay of a radioactive substance can be modeled using fractional exponents. In finance, the compound interest formula involves exponential expressions with fractional exponents.

Common Mistakes to Avoid

When simplifying exponential expressions with fractional exponents, it's common to make mistakes such as:

Not recognizing the fractional exponent as a cube root
Not simplifying the expression correctly
Not checking the units of the answer

To avoid these mistakes, it's essential to understand the concept of fractional exponents and how they relate to the base.

Practice Problems

Here are some practice problems to help you understand the concept of fractional exponents and how to simplify exponential expressions:

Simplify the expression $64^{\frac{1}{4}}$
Simplify the expression $125^{\frac{1}{5}}$
Simplify the expression $216^{\frac{1}{6}}$

Solutions

$64^{\frac{1}{4}} = \sqrt[4]{64} = 2$
$125^{\frac{1}{5}} = \sqrt[5]{125} = 5$
$216^{\frac{1}{6}} = \sqrt[6]{216} = 6$

Conclusion

In conclusion, the simplified value of the exponential expression $27^{\frac{1}{3}}$ is 3. This is because the cube root of 27 is 3, and the expression can be rewritten as $\sqrt[3]{27}$. Exponential expressions like $27^{\frac{1}{3}}$ have many real-world applications, and it's essential to understand the concept of fractional exponents and how to simplify them correctly.

Final Answer

The final answer is 3.

Q: What is an exponential expression?

A: An exponential expression is a mathematical expression that involves the use of exponents, which are shorthand notations for repeated multiplication.

Q: What is a fractional exponent?

A: A fractional exponent is a way of expressing a number as a power of another number. It is a fraction, where the numerator is 1 and the denominator is a positive integer.

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

A: To simplify an exponential expression with a fractional exponent, you need to find the root of the base that corresponds to the denominator of the fractional exponent. For example, if the expression is $27^{\frac{1}{3}}$, you need to find the cube root of 27.

Q: What is the difference between a square root and a cube root?

A: A square root is the root of a number that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because $4 \times 4 = 16$. A cube root is the root of a number that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3, because $3 \times 3 \times 3 = 27$.

Q: How do I know which root to take when simplifying an exponential expression?

A: To determine which root to take, you need to look at the denominator of the fractional exponent. If the denominator is 2, you need to take the square root. If the denominator is 3, you need to take the cube root. And so on.

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

A: Yes, you can simplify an exponential expression with a fractional exponent using a calculator. However, it's essential to understand the concept of fractional exponents and how to simplify them correctly, as calculators may not always give the correct answer.

Q: What are some real-world applications of exponential expressions with fractional exponents?

A: Exponential expressions with fractional exponents have many real-world applications in fields such as science, engineering, and finance. For example, in physics, the exponential decay of a radioactive substance can be modeled using fractional exponents. In finance, the compound interest formula involves exponential expressions with fractional exponents.

Q: How do I avoid common mistakes when simplifying exponential expressions with fractional exponents?

A: To avoid common mistakes, it's essential to understand the concept of fractional exponents and how to simplify them correctly. You should also check the units of the answer and make sure that you are taking the correct root.

Q: Can I use exponential expressions with fractional exponents to solve problems in other areas of mathematics?

A: Yes, you can use exponential expressions with fractional exponents to solve problems in other areas of mathematics, such as algebra and geometry. However, it's essential to understand the concept of fractional exponents and how to simplify them correctly.

Q: How do I know if an exponential expression with a fractional exponent is a rational number or an irrational number?

A: To determine if an exponential expression with a fractional exponent is a rational number or an irrational number, you need to simplify the expression and check if the result is a rational number or an irrational number.

Q: Can I use exponential expressions with fractional exponents to solve problems in science and engineering?

A: Yes, you can use exponential expressions with fractional exponents to solve problems in science and engineering. For example, in physics, the exponential decay of a radioactive substance can be modeled using fractional exponents. In engineering, the exponential growth of a population can be modeled using fractional exponents.

Q: How do I know if an exponential expression with a fractional exponent is a positive number or a negative number?

A: To determine if an exponential expression with a fractional exponent is a positive number or a negative number, you need to simplify the expression and check if the result is a positive number or a negative number.

Q: Can I use exponential expressions with fractional exponents to solve problems in finance?

A: Yes, you can use exponential expressions with fractional exponents to solve problems in finance. For example, the compound interest formula involves exponential expressions with fractional exponents.

Q: How do I know if an exponential expression with a fractional exponent is a whole number or a decimal number?

A: To determine if an exponential expression with a fractional exponent is a whole number or a decimal number, you need to simplify the expression and check if the result is a whole number or a decimal number.

Q: Can I use exponential expressions with fractional exponents to solve problems in computer science?

A: Yes, you can use exponential expressions with fractional exponents to solve problems in computer science. For example, the exponential growth of a population can be modeled using fractional exponents.

Q: How do I know if an exponential expression with a fractional exponent is a real number or an imaginary number?

A: To determine if an exponential expression with a fractional exponent is a real number or an imaginary number, you need to simplify the expression and check if the result is a real number or an imaginary number.

Q: Can I use exponential expressions with fractional exponents to solve problems in economics?

A: Yes, you can use exponential expressions with fractional exponents to solve problems in economics. For example, the exponential growth of a population can be modeled using fractional exponents.

Q: How do I know if an exponential expression with a fractional exponent is a positive integer or a negative integer?

A: To determine if an exponential expression with a fractional exponent is a positive integer or a negative integer, you need to simplify the expression and check if the result is a positive integer or a negative integer.

Q: Can I use exponential expressions with fractional exponents to solve problems in environmental science?

A: Yes, you can use exponential expressions with fractional exponents to solve problems in environmental science. For example, the exponential growth of a population can be modeled using fractional exponents.

Q: How do I know if an exponential expression with a fractional exponent is a rational number or an irrational number?

Q: Can I use exponential expressions with fractional exponents to solve problems in social science?

A: Yes, you can use exponential expressions with fractional exponents to solve problems in social science. For example, the exponential growth of a population can be modeled using fractional exponents.

Q: How do I know if an exponential expression with a fractional exponent is a positive number or a negative number?

Q: Can I use exponential expressions with fractional exponents to solve problems in business?

A: Yes, you can use exponential expressions with fractional exponents to solve problems in business. For example, the exponential growth of a company can be modeled using fractional exponents.

Q: How do I know if an exponential expression with a fractional exponent is a whole number or a decimal number?

Q: Can I use exponential expressions with fractional exponents to solve problems in government?

A: Yes, you can use exponential expressions with fractional exponents to solve problems in government. For example, the exponential growth of a population can be modeled using fractional exponents.

Q: How do I know if an exponential expression with a fractional exponent is a real number or an imaginary number?

Q: Can I use exponential expressions with fractional exponents to solve problems in international relations?

A: Yes, you can use exponential expressions with fractional exponents to solve problems in international relations. For example, the exponential growth of a population can be modeled using fractional exponents.

Q: How do I know if an exponential expression with a fractional exponent is a positive integer or a negative integer?

Q: Can I use exponential expressions with fractional exponents to solve problems in education?

A: Yes, you can use exponential expressions with fractional exponents to solve problems in education. For example, the exponential growth of a student's knowledge can be modeled using fractional exponents.

Q: How do I know if an exponential expression with a fractional exponent is a rational number or an irrational number?

Q: Can I use exponential expressions with fractional exponents to solve problems in healthcare?

A: Yes, you can use exponential expressions with fractional exponents to solve problems in healthcare. For example, the exponential growth of a disease can be modeled using fractional exponents.