Which Of The Following Is Equal To The Fraction Below?$\left(\frac{4}{5}\right)^6$A. $6 \cdot\left(\frac{4}{5}\right$\]B. $\frac{4^6}{5^6}$C. $\frac{4^6}{5}$D. $\frac{24}{30}$

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Which of the Following is Equal to the Fraction Below?

Understanding the Problem

When dealing with exponents and fractions, it's essential to understand the rules and properties that govern their behavior. In this problem, we're given the expression (45)6\left(\frac{4}{5}\right)^6 and asked to determine which of the provided options is equal to it.

The Power of a Fraction

To tackle this problem, we need to recall the rule for raising a fraction to a power. When a fraction is raised to a power, both the numerator and the denominator are raised to that power. In other words, (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}.

Applying the Rule

Using this rule, we can rewrite the given expression as (45)6=4656\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}. This is a direct application of the rule, where both the numerator and the denominator are raised to the power of 6.

Evaluating the Options

Now that we have the correct expression, let's evaluate the options provided:

A. 6 \cdot\left(\frac{4}{5}\right] - This option is incorrect because it doesn't follow the rule for raising a fraction to a power.

B. 4656\frac{4^6}{5^6} - This option is correct because it matches the expression we derived using the rule.

C. 465\frac{4^6}{5} - This option is incorrect because it doesn't follow the rule for raising a fraction to a power.

D. 2430\frac{24}{30} - This option is incorrect because it doesn't match the expression we derived using the rule.

Conclusion

In conclusion, the correct answer is option B, 4656\frac{4^6}{5^6}. This is because it follows the rule for raising a fraction to a power and matches the expression we derived using this rule.

Additional Tips and Tricks

When dealing with exponents and fractions, it's essential to remember the following tips and tricks:

  • When raising a fraction to a power, both the numerator and the denominator are raised to that power.
  • Use the rule (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} to simplify expressions involving exponents and fractions.
  • Be careful when evaluating options, as they may not follow the rules and properties of exponents and fractions.

Common Mistakes to Avoid

When dealing with exponents and fractions, some common mistakes to avoid include:

  • Not following the rule for raising a fraction to a power.
  • Not simplifying expressions using the rule (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}.
  • Not being careful when evaluating options, as they may not follow the rules and properties of exponents and fractions.

Real-World Applications

Understanding the rules and properties of exponents and fractions has numerous real-world applications, including:

  • Calculating interest rates and investments.
  • Determining the area and volume of shapes and objects.
  • Solving problems involving growth and decay.

Conclusion

In conclusion, the correct answer is option B, 4656\frac{4^6}{5^6}. This is because it follows the rule for raising a fraction to a power and matches the expression we derived using this rule. By understanding the rules and properties of exponents and fractions, we can solve a wide range of problems and make informed decisions in various real-world applications.

Final Thoughts

When dealing with exponents and fractions, it's essential to remember the rules and properties that govern their behavior. By following these rules and being careful when evaluating options, we can ensure that our calculations are accurate and our conclusions are correct.
Q&A: Exponents and Fractions

Understanding Exponents and Fractions

Exponents and fractions are fundamental concepts in mathematics that are used to represent and manipulate numbers. In this article, we'll answer some common questions about exponents and fractions, and provide tips and tricks for simplifying expressions and solving problems.

Q: What is an exponent?

A: An exponent is a small number that is written above and to the right of a base number. It represents the number of times the base number is multiplied by itself. For example, in the expression 232^3, the exponent 3 represents the number of times 2 is multiplied by itself: 23=2×2×2=82^3 = 2 \times 2 \times 2 = 8.

Q: What is a fraction?

A: A fraction is a way of representing a part of a whole. It consists of a numerator (the top number) and a denominator (the bottom number). For example, in the fraction 12\frac{1}{2}, the numerator 1 represents one part of the whole, and the denominator 2 represents the total number of parts.

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

A: To simplify an expression with exponents and fractions, you can use the following steps:

  1. Simplify the fraction by dividing the numerator and denominator by their greatest common divisor (GCD).
  2. Apply the rule for raising a fraction to a power: (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}.
  3. Simplify the resulting expression by combining like terms.

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

A: A positive exponent represents the number of times the base number is multiplied by itself, while a negative exponent represents the reciprocal of the base number raised to the positive exponent. For example, in the expression 2−32^{-3}, the negative exponent represents the reciprocal of 2 raised to the power of 3: 2−3=123=182^{-3} = \frac{1}{2^3} = \frac{1}{8}.

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

A: To evaluate an expression with exponents and fractions, you can use the following steps:

  1. Simplify the fraction by dividing the numerator and denominator by their GCD.
  2. Apply the rule for raising a fraction to a power: (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}.
  3. Simplify the resulting expression by combining like terms.
  4. Evaluate the expression by performing the necessary calculations.

Q: What are some common mistakes to avoid when working with exponents and fractions?

A: Some common mistakes to avoid when working with exponents and fractions include:

  • Not following the rule for raising a fraction to a power.
  • Not simplifying expressions using the rule (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}.
  • Not being careful when evaluating expressions, as they may involve complex calculations.

Q: How do I apply exponents and fractions in real-world applications?

A: Exponents and fractions are used in a wide range of real-world applications, including:

  • Calculating interest rates and investments.
  • Determining the area and volume of shapes and objects.
  • Solving problems involving growth and decay.

Conclusion

In conclusion, exponents and fractions are fundamental concepts in mathematics that are used to represent and manipulate numbers. By understanding the rules and properties of exponents and fractions, we can simplify expressions and solve problems with ease. Remember to follow the rule for raising a fraction to a power, simplify expressions using the rule (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}, and be careful when evaluating expressions.

Final Thoughts

Exponents and fractions are powerful tools that can be used to solve a wide range of problems. By understanding the rules and properties of exponents and fractions, we can make informed decisions and solve complex problems with ease. Remember to practice regularly and seek help when needed to become proficient in working with exponents and fractions.