Evaluate The Expression: $\left(\frac{4 \cdot 3}{5}\right)^5$

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Introduction

In mathematics, expressions involving exponents and fractions can be challenging to evaluate. The given expression, (4⋅35)5\left(\frac{4 \cdot 3}{5}\right)^5, requires us to apply the rules of exponentiation and fraction arithmetic to simplify it. In this article, we will break down the expression step by step, using the properties of exponents and fractions to arrive at the final result.

Understanding Exponents and Fractions

Before we dive into the evaluation of the expression, let's review the basic rules of exponents and fractions.

  • Exponents: An exponent is a small number that is written above and to the right of a larger number, indicating how many times the larger number should be multiplied by itself. For example, aba^b means aa multiplied by itself bb times.
  • Fractions: A fraction is a way of expressing a part of a whole. It consists of a numerator (the top number) and a denominator (the bottom number). For example, ab\frac{a}{b} means aa divided by bb.

Evaluating the Expression

Now that we have a basic understanding of exponents and fractions, let's evaluate the given expression.

(4⋅35)5\left(\frac{4 \cdot 3}{5}\right)^5

To evaluate this expression, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate the expression inside the parentheses first.
  2. Exponents: Evaluate any exponents next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Step 1: Evaluate the Expression Inside the Parentheses

The expression inside the parentheses is 4⋅35\frac{4 \cdot 3}{5}. To evaluate this expression, we need to follow the order of operations:

  1. Multiplication: Multiply 4 and 3.
  2. Division: Divide the result by 5.

4⋅35=125\frac{4 \cdot 3}{5} = \frac{12}{5}

Step 2: Evaluate the Exponent

Now that we have evaluated the expression inside the parentheses, we can evaluate the exponent:

(125)5\left(\frac{12}{5}\right)^5

To evaluate this expression, we need to multiply 125\frac{12}{5} by itself 5 times:

(125)5=125⋅125⋅125⋅125⋅125\left(\frac{12}{5}\right)^5 = \frac{12}{5} \cdot \frac{12}{5} \cdot \frac{12}{5} \cdot \frac{12}{5} \cdot \frac{12}{5}

Step 3: Simplify the Expression

Now that we have multiplied 125\frac{12}{5} by itself 5 times, we can simplify the expression:

125⋅125⋅125⋅125⋅125=12555\frac{12}{5} \cdot \frac{12}{5} \cdot \frac{12}{5} \cdot \frac{12}{5} \cdot \frac{12}{5} = \frac{12^5}{5^5}

Step 4: Evaluate the Exponent

Now that we have simplified the expression, we can evaluate the exponent:

12555\frac{12^5}{5^5}

To evaluate this expression, we need to multiply 12 by itself 5 times and divide the result by 5 raised to the power of 5:

125=24883212^5 = 248832

55=31255^5 = 3125

12555=2488323125\frac{12^5}{5^5} = \frac{248832}{3125}

Step 5: Simplify the Fraction

Now that we have evaluated the exponent, we can simplify the fraction:

2488323125=79.54\frac{248832}{3125} = 79.54

Conclusion

In this article, we evaluated the expression (4⋅35)5\left(\frac{4 \cdot 3}{5}\right)^5 using the properties of exponents and fractions. We broke down the expression step by step, following the order of operations and simplifying the fraction at the end. The final result is 2488323125\frac{248832}{3125}, which can be simplified to 79.54.

Frequently Asked Questions

  • What is the order of operations?
    • The order of operations is a set of rules that dictate the order in which mathematical operations should be performed. The acronym PEMDAS is commonly used to remember the order of operations:
      • Parentheses: Evaluate expressions inside parentheses first.
      • Exponents: Evaluate any exponents next.
      • Multiplication and Division: Evaluate any multiplication and division operations from left to right.
      • Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
  • How do I evaluate an expression with exponents and fractions?
    • To evaluate an expression with exponents and fractions, follow the order of operations:
      1. Evaluate any expressions inside parentheses first.
      2. Evaluate any exponents next.
      3. Evaluate any multiplication and division operations from left to right.
      4. Finally, evaluate any addition and subtraction operations from left to right.
  • What is the difference between a fraction and a decimal?
    • A fraction is a way of expressing a part of a whole, consisting of a numerator (the top number) and a denominator (the bottom number). A decimal is a way of expressing a number as a sum of powers of 10. For example, the fraction 12\frac{1}{2} is equivalent to the decimal 0.5.

Introduction

In our previous article, we evaluated the expression (4⋅35)5\left(\frac{4 \cdot 3}{5}\right)^5 using the properties of exponents and fractions. In this article, we will answer some frequently asked questions related to evaluating expressions with exponents and fractions.

Q&A

Q: What is the order of operations?

A: The order of operations is a set of rules that dictate the order in which mathematical operations should be performed. The acronym PEMDAS is commonly used to remember the order of operations:

  • Parentheses: Evaluate expressions inside parentheses first.
  • Exponents: Evaluate any exponents next.
  • Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  • Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I evaluate an expression with exponents and fractions?

A: To evaluate an expression with exponents and fractions, follow the order of operations:

  1. Evaluate any expressions inside parentheses first.
  2. Evaluate any exponents next.
  3. Evaluate any multiplication and division operations from left to right.
  4. Finally, evaluate any addition and subtraction operations from left to right.

Q: What is the difference between a fraction and a decimal?

A: A fraction is a way of expressing a part of a whole, consisting of a numerator (the top number) and a denominator (the bottom number). A decimal is a way of expressing a number as a sum of powers of 10. For example, the fraction 12\frac{1}{2} is equivalent to the decimal 0.5.

Q: How do I simplify a fraction?

A: To simplify a fraction, find the greatest common divisor (GCD) of the numerator and denominator, and divide both numbers by the GCD. For example, to simplify the fraction 1216\frac{12}{16}, find the GCD of 12 and 16, which is 4. Then, divide both numbers by 4 to get 34\frac{3}{4}.

Q: What is the rule for multiplying fractions?

A: When multiplying fractions, multiply the numerators together and multiply the denominators together. For example, to multiply the fractions 12\frac{1}{2} and 34\frac{3}{4}, multiply the numerators together to get 3, and multiply the denominators together to get 8. The result is 38\frac{3}{8}.

Q: What is the rule for dividing fractions?

A: When dividing fractions, invert the second fraction (i.e., flip the numerator and denominator) and multiply. For example, to divide the fractions 12\frac{1}{2} and 34\frac{3}{4}, invert the second fraction to get 43\frac{4}{3}, and multiply to get 46\frac{4}{6}, which simplifies to 23\frac{2}{3}.

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

A: To evaluate an expression with multiple exponents, follow the order of operations:

  1. Evaluate any expressions inside parentheses first.
  2. Evaluate any exponents next.
  3. Evaluate any multiplication and division operations from left to right.
  4. Finally, evaluate any addition and subtraction operations from left to right.

For example, to evaluate the expression (23)4(2^3)^4, follow the order of operations:

  1. Evaluate the expression inside the parentheses: 23=82^3 = 8
  2. Evaluate the exponent: 84=40968^4 = 4096

Q: What is the rule for evaluating expressions with negative exponents?

A: When evaluating an expression with a negative exponent, rewrite the expression with a positive exponent by moving the base to the other side of the fraction. For example, to evaluate the expression 2−32^{-3}, rewrite the expression as 123\frac{1}{2^3}, which simplifies to 18\frac{1}{8}.

Conclusion

In this article, we answered some frequently asked questions related to evaluating expressions with exponents and fractions. We covered topics such as the order of operations, simplifying fractions, multiplying and dividing fractions, and evaluating expressions with multiple exponents and negative exponents. By following the order of operations and applying the rules for exponents and fractions, you can evaluate complex expressions with ease.

Frequently Asked Questions

  • What is the order of operations?
    • The order of operations is a set of rules that dictate the order in which mathematical operations should be performed. The acronym PEMDAS is commonly used to remember the order of operations:
      • Parentheses: Evaluate expressions inside parentheses first.
      • Exponents: Evaluate any exponents next.
      • Multiplication and Division: Evaluate any multiplication and division operations from left to right.
      • Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
  • How do I evaluate an expression with exponents and fractions?
    • To evaluate an expression with exponents and fractions, follow the order of operations:
      1. Evaluate any expressions inside parentheses first.
      2. Evaluate any exponents next.
      3. Evaluate any multiplication and division operations from left to right.
      4. Finally, evaluate any addition and subtraction operations from left to right.
  • What is the difference between a fraction and a decimal?
    • A fraction is a way of expressing a part of a whole, consisting of a numerator (the top number) and a denominator (the bottom number). A decimal is a way of expressing a number as a sum of powers of 10. For example, the fraction 12\frac{1}{2} is equivalent to the decimal 0.5.
  • How do I simplify a fraction?
    • To simplify a fraction, find the greatest common divisor (GCD) of the numerator and denominator, and divide both numbers by the GCD. For example, to simplify the fraction 1216\frac{12}{16}, find the GCD of 12 and 16, which is 4. Then, divide both numbers by 4 to get 34\frac{3}{4}.
  • What is the rule for multiplying fractions?
    • When multiplying fractions, multiply the numerators together and multiply the denominators together. For example, to multiply the fractions 12\frac{1}{2} and 34\frac{3}{4}, multiply the numerators together to get 3, and multiply the denominators together to get 8. The result is 38\frac{3}{8}.
  • What is the rule for dividing fractions?
    • When dividing fractions, invert the second fraction (i.e., flip the numerator and denominator) and multiply. For example, to divide the fractions 12\frac{1}{2} and 34\frac{3}{4}, invert the second fraction to get 43\frac{4}{3}, and multiply to get 46\frac{4}{6}, which simplifies to 23\frac{2}{3}.
  • How do I evaluate an expression with multiple exponents?
    • To evaluate an expression with multiple exponents, follow the order of operations:
      1. Evaluate any expressions inside parentheses first.
      2. Evaluate any exponents next.
      3. Evaluate any multiplication and division operations from left to right.
      4. Finally, evaluate any addition and subtraction operations from left to right.
  • What is the rule for evaluating expressions with negative exponents?
    • When evaluating an expression with a negative exponent, rewrite the expression with a positive exponent by moving the base to the other side of the fraction. For example, to evaluate the expression 2−32^{-3}, rewrite the expression as 123\frac{1}{2^3}, which simplifies to 18\frac{1}{8}.