Evaluate The Expression:$\[ \frac{27^2}{27^{\frac{4}{3}}} \\]

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Introduction

In mathematics, exponents are a fundamental concept used to represent repeated multiplication of a number. When evaluating expressions with exponents, it's essential to understand the rules and properties of exponents to simplify and solve complex expressions. In this article, we will evaluate the expression 2722743\frac{27^2}{27^{\frac{4}{3}}} and explore the properties of exponents that make this evaluation possible.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication of a number. For example, 232^3 means 22 multiplied by itself 33 times, which is equal to 88. Exponents can be positive, negative, or fractional, and they follow specific rules and properties that are essential to understand when evaluating expressions.

Properties of Exponents

There are several properties of exponents that are crucial to understand when evaluating expressions. These properties include:

  • Product of Powers: When multiplying two numbers with the same base, we can add their exponents. For example, 23â‹…24=23+4=272^3 \cdot 2^4 = 2^{3+4} = 2^7.
  • Power of a Power: When raising a power to another power, we can multiply the exponents. For example, (23)4=23â‹…4=212(2^3)^4 = 2^{3\cdot4} = 2^{12}.
  • Quotient of Powers: When dividing two numbers with the same base, we can subtract their exponents. For example, 2322=23−2=21\frac{2^3}{2^2} = 2^{3-2} = 2^1.

Evaluating the Expression

Now that we have a solid understanding of exponents and their properties, let's evaluate the expression 2722743\frac{27^2}{27^{\frac{4}{3}}}. To do this, we can use the quotient of powers property, which states that when dividing two numbers with the same base, we can subtract their exponents.

2722743=272−43\frac{27^2}{27^{\frac{4}{3}}} = 27^{2-\frac{4}{3}}

To simplify the exponent, we can find a common denominator, which is 33. This gives us:

2−43=63−43=232-\frac{4}{3} = \frac{6}{3} - \frac{4}{3} = \frac{2}{3}

So, the expression becomes:

272327^{\frac{2}{3}}

Simplifying the Expression

Now that we have simplified the exponent, we can evaluate the expression by raising 2727 to the power of 23\frac{2}{3}. To do this, we can use the fact that 27=3327 = 3^3. This gives us:

2723=(33)2327^{\frac{2}{3}} = (3^3)^{\frac{2}{3}}

Using the power of a power property, we can multiply the exponents:

(33)23=33â‹…23=32(3^3)^{\frac{2}{3}} = 3^{3\cdot\frac{2}{3}} = 3^2

So, the expression simplifies to:

32=93^2 = 9

Conclusion

In this article, we evaluated the expression 2722743\frac{27^2}{27^{\frac{4}{3}}} using the properties of exponents. We used the quotient of powers property to simplify the exponent and then evaluated the expression by raising 2727 to the power of 23\frac{2}{3}. This gave us a final answer of 99. By understanding the properties of exponents and how to apply them, we can simplify complex expressions and solve a wide range of mathematical problems.

Common Mistakes to Avoid

When evaluating expressions with exponents, there are several common mistakes to avoid. These include:

  • Forgetting to simplify the exponent: When simplifying an expression with exponents, it's essential to simplify the exponent before evaluating the expression.
  • Not using the correct property of exponents: When evaluating an expression with exponents, it's essential to use the correct property of exponents. For example, when dividing two numbers with the same base, we can subtract their exponents.
  • Not checking the final answer: When evaluating an expression with exponents, it's essential to check the final answer to ensure that it's correct.

Real-World Applications

Exponents have a wide range of real-world applications, including:

  • Finance: Exponents are used to calculate compound interest and investment returns.
  • Science: Exponents are used to describe the growth and decay of populations, chemical reactions, and other scientific phenomena.
  • Engineering: Exponents are used to describe the behavior of electrical circuits, mechanical systems, and other engineering applications.

Practice Problems

To practice evaluating expressions with exponents, try the following problems:

  • 2322\frac{2^3}{2^2}
  • 3432\frac{3^4}{3^2}
  • 42432\frac{4^2}{4^{\frac{3}{2}}}

Conclusion

Introduction

In our previous article, we evaluated the expression 2722743\frac{27^2}{27^{\frac{4}{3}}} using the properties of exponents. We used the quotient of powers property to simplify the exponent and then evaluated the expression by raising 2727 to the power of 23\frac{2}{3}. This gave us a final answer of 99. In this article, we will answer some common questions about evaluating expressions with exponents.

Q: What is the quotient of powers property?

A: The quotient of powers property states that when dividing two numbers with the same base, we can subtract their exponents. For example, 2322=23−2=21\frac{2^3}{2^2} = 2^{3-2} = 2^1.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you need to follow these steps:

  1. Identify the base and the exponents in the expression.
  2. Use the quotient of powers property to simplify the exponent.
  3. Evaluate the expression by raising the base to the simplified exponent.

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

A: A positive exponent represents repeated multiplication of a number, while a negative exponent represents repeated division of a number. For example, 23=2⋅2⋅22^3 = 2 \cdot 2 \cdot 2 and 2−3=12⋅2⋅22^{-3} = \frac{1}{2 \cdot 2 \cdot 2}.

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

A: To evaluate an expression with a fractional exponent, you need to follow these steps:

  1. Identify the base and the fractional exponent in the expression.
  2. Simplify the fractional exponent by finding a common denominator.
  3. Evaluate the expression by raising the base to the simplified exponent.

Q: What is the product of powers property?

A: The product of powers property states that when multiplying two numbers with the same base, we can add their exponents. For example, 23â‹…24=23+4=272^3 \cdot 2^4 = 2^{3+4} = 2^7.

Q: How do I evaluate an expression with multiple exponents?

A: To evaluate an expression with multiple exponents, you need to follow these steps:

  1. Identify the base and the exponents in the expression.
  2. Use the product of powers property to simplify the exponents.
  3. Evaluate the expression by raising the base to the simplified exponent.

Q: What is the power of a power property?

A: The power of a power property states that when raising a power to another power, we can multiply the exponents. For example, (23)4=23â‹…4=212(2^3)^4 = 2^{3\cdot4} = 2^{12}.

Q: How do I evaluate an expression with a power of a power?

A: To evaluate an expression with a power of a power, you need to follow these steps:

  1. Identify the base and the exponents in the expression.
  2. Use the power of a power property to simplify the exponents.
  3. Evaluate the expression by raising the base to the simplified exponent.

Conclusion

In this article, we answered some common questions about evaluating expressions with exponents. We covered topics such as the quotient of powers property, simplifying expressions with exponents, and evaluating expressions with fractional exponents. By understanding these concepts and how to apply them, you can simplify complex expressions and solve a wide range of mathematical problems.

Practice Problems

To practice evaluating expressions with exponents, try the following problems:

  • 2322\frac{2^3}{2^2}
  • 3432\frac{3^4}{3^2}
  • 42432\frac{4^2}{4^{\frac{3}{2}}}
  • (23)4(2^3)^4
  • 23â‹…42^{3\cdot4}

Real-World Applications

Exponents have a wide range of real-world applications, including:

  • Finance: Exponents are used to calculate compound interest and investment returns.
  • Science: Exponents are used to describe the growth and decay of populations, chemical reactions, and other scientific phenomena.
  • Engineering: Exponents are used to describe the behavior of electrical circuits, mechanical systems, and other engineering applications.

Common Mistakes to Avoid

When evaluating expressions with exponents, there are several common mistakes to avoid. These include:

  • Forgetting to simplify the exponent: When simplifying an expression with exponents, it's essential to simplify the exponent before evaluating the expression.
  • Not using the correct property of exponents: When evaluating an expression with exponents, it's essential to use the correct property of exponents. For example, when dividing two numbers with the same base, we can subtract their exponents.
  • Not checking the final answer: When evaluating an expression with exponents, it's essential to check the final answer to ensure that it's correct.