What Is The Simplified Value Of The Exponential Expression $16^{\frac{1}{4}}$?A. $\frac{1}{2}$ B. $ 1 4 \frac{1}{4} 4 1 ​ [/tex] C. 2 D. 4

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Introduction

Exponential expressions 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 the exponential expression $16^{\frac{1}{4}}$. We will explore the concept of fractional exponents, the properties of exponents, and how to apply them to simplify the given expression.

What are Exponential Expressions?

Exponential expressions are mathematical expressions that involve a base number raised to a power. The base number is the number that is being raised to the power, and the exponent is the power to which the base number is raised. For example, in the expression $2^3$, the base number is 2 and the exponent is 3.

Fractional Exponents

Fractional exponents are exponents that are expressed as fractions. They can be written in the form $a^{\frac{m}{n}}$, where a is the base number, m is the numerator, and n is the denominator. Fractional exponents can be used to simplify expressions and to represent repeated multiplication.

Simplifying $16^{\frac{1}{4}}$

To simplify the expression $16^{\frac{1}{4}}$, we need to understand the concept of fractional exponents and the properties of exponents. The expression $16^{\frac{1}{4}}$ can be rewritten as $(24){\frac{1}{4}}$. Using the property of exponents that states $(am)n = a^{mn}$, we can simplify the expression as follows:

(24)14=24×14=21=2(2^4)^{\frac{1}{4}} = 2^{4 \times \frac{1}{4}} = 2^1 = 2

Therefore, the simplified value of the exponential expression $16^{\frac{1}{4}}$ is 2.

Conclusion

In this article, we have explored the concept of exponential expressions and how to simplify them. We have focused on the expression $16^{\frac{1}{4}}$ and have shown how to simplify it using the properties of exponents. The simplified value of the expression is 2. We hope that this article has provided a clear understanding of how to simplify exponential expressions and has been helpful in solving mathematical problems.

Properties of Exponents

Exponents have several properties that can be used to simplify expressions. Some of the key properties of exponents are:

  • Product of Powers Property: $(am)(an) = a^{m+n}$
  • Power of a Power Property: $(am)n = a^{mn}$
  • Power of a Product Property: $(ab)^m = ambm$
  • Zero Exponent Property: $a^0 = 1$
  • Negative Exponent Property: $a^{-m} = \frac{1}{a^m}$

Examples of Simplifying Exponential Expressions

Here are some examples of simplifying exponential expressions using the properties of exponents:

  • 23×24=23+4=272^3 \times 2^4 = 2^{3+4} = 2^7

  • (23)4=23×4=212(2^3)^4 = 2^{3 \times 4} = 2^{12}

  • (2×3)4=24×34(2 \times 3)^4 = 2^4 \times 3^4

  • 20=12^0 = 1

  • 2−3=123=182^{-3} = \frac{1}{2^3} = \frac{1}{8}

Applications of Exponential Expressions

Exponential expressions have numerous applications in mathematics and other fields. Some of the key applications of exponential expressions include:

  • Finance: Exponential expressions are used to calculate interest rates and compound interest.
  • Science: Exponential expressions are used to model population growth and decay.
  • Engineering: Exponential expressions are used to design and analyze electrical circuits.
  • Computer Science: Exponential expressions are used to represent and manipulate data.

Conclusion

Q: What is an exponential expression?

A: An exponential expression is a mathematical expression that involves a base number raised to a power. The base number is the number that is being raised to the power, and the exponent is the power to which the base number is raised.

Q: What is a fractional exponent?

A: A fractional exponent is an exponent that is expressed as a fraction. It can be written in the form $a^{\frac{m}{n}}$, where a is the base number, m is the numerator, and n is the denominator.

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 understand the concept of fractional exponents and the properties of exponents. You can use the property of exponents that states $(am)n = a^{mn}$ to simplify the expression.

Q: What is the simplified value of the expression $16^{\frac{1}{4}}$?

A: The simplified value of the expression $16^{\frac{1}{4}}$ is 2. This is because $16^{\frac{1}{4}} = (24){\frac{1}{4}} = 2^{4 \times \frac{1}{4}} = 2^1 = 2$.

Q: What are some common properties of exponents?

A: Some common properties of exponents include:

  • Product of Powers Property: $(am)(an) = a^{m+n}$
  • Power of a Power Property: $(am)n = a^{mn}$
  • Power of a Product Property: $(ab)^m = ambm$
  • Zero Exponent Property: $a^0 = 1$
  • Negative Exponent Property: $a^{-m} = \frac{1}{a^m}$

Q: How do I apply the properties of exponents to simplify an expression?

A: To apply the properties of exponents to simplify an expression, you need to identify the properties that can be used to simplify the expression. For example, if you have an expression like $2^3 \times 2^4$, you can use the product of powers property to simplify it as $2^{3+4} = 2^7$.

Q: What are some examples of simplifying exponential expressions using the properties of exponents?

A: Here are some examples of simplifying exponential expressions using the properties of exponents:

  • 23×24=23+4=272^3 \times 2^4 = 2^{3+4} = 2^7

  • (23)4=23×4=212(2^3)^4 = 2^{3 \times 4} = 2^{12}

  • (2×3)4=24×34(2 \times 3)^4 = 2^4 \times 3^4

  • 20=12^0 = 1

  • 2−3=123=182^{-3} = \frac{1}{2^3} = \frac{1}{8}

Q: What are some applications of exponential expressions in real-life situations?

A: Exponential expressions have numerous applications in real-life situations, including:

  • Finance: Exponential expressions are used to calculate interest rates and compound interest.
  • Science: Exponential expressions are used to model population growth and decay.
  • Engineering: Exponential expressions are used to design and analyze electrical circuits.
  • Computer Science: Exponential expressions are used to represent and manipulate data.

Q: How can I use exponential expressions to solve problems in mathematics and other fields?

A: You can use exponential expressions to solve problems in mathematics and other fields by applying the properties of exponents and using the concepts of fractional exponents. For example, you can use exponential expressions to calculate interest rates, model population growth, and design electrical circuits.