Simplify { 2^{4 / 5} \div \frac{2}{3} \times 4 \frac{1}{5} .}

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Simplify { 2^{4 / 5} \div \frac{2}{3} \times 4 \frac{1}{5} .}

In this article, we will simplify the given mathematical expression: $ 2^{4 / 5} \div \frac{2}{3} \times 4 \frac{1}{5} $. This expression involves exponents, fractions, and mixed numbers, making it a challenging problem to solve. We will break down the expression step by step, using the order of operations and properties of exponents to simplify it.

Step 1: Evaluate the Exponent

The first step is to evaluate the exponent $ 2^{4 / 5} ∗∗.Todothis,weneedtounderstandthattheexponent∗∗**. To do this, we need to understand that the exponent ** 4 / 5 ∗∗representsafractionalexponent.Afractionalexponentcanbewrittenas∗∗** represents a fractional exponent. A fractional exponent can be written as ** a^{m/n} = \sqrt[n]{a^m} ∗∗,where∗∗**, where ** a ∗∗isthebaseand∗∗** is the base and ** m/n $ is the exponent.

In this case, the base is $ 2 ∗∗andtheexponentis∗∗** and the exponent is ** 4 / 5 ∗∗.Wecanrewritetheexpressionas∗∗**. We can rewrite the expression as ** \sqrt[5]{2^4} $.

Step 2: Simplify the Fractional Exponent

Now, we need to simplify the fractional exponent $ \sqrt[5]{2^4} ∗∗.Todothis,wecanevaluatetheexponent∗∗**. To do this, we can evaluate the exponent ** 2^4 ∗∗first.∗∗** first. ** 2^4 = 16 ∗∗,sotheexpressionbecomes∗∗**, so the expression becomes ** \sqrt[5]{16} $.

Step 3: Simplify the Radical

Next, we need to simplify the radical $ \sqrt[5]{16} ∗∗.Todothis,wecanfindthefifthrootof∗∗**. To do this, we can find the fifth root of ** 16 ∗∗.Thefifthrootof∗∗**. The fifth root of ** 16 ∗∗is∗∗** is ** 2 ∗∗,sotheexpressionbecomes∗∗**, so the expression becomes ** 2 $.

Step 4: Simplify the Fraction

Now, we need to simplify the fraction $ \frac{2}{3} $. This fraction is already in its simplest form, so we can leave it as is.

Step 5: Simplify the Mixed Number

Next, we need to simplify the mixed number $ 4 \frac{1}{5} ∗∗.Todothis,wecanconvertthemixednumbertoanimproperfraction.Themixednumber∗∗**. To do this, we can convert the mixed number to an improper fraction. The mixed number ** 4 \frac{1}{5} ∗∗isequalto∗∗** is equal to ** \frac{21}{5} $.

Step 6: Multiply the Fractions

Now, we need to multiply the fractions $ 2 ∗∗,∗∗**, ** \frac{2}{3} ∗∗,and∗∗**, and ** \frac{21}{5} $. To do this, we can multiply the numerators and denominators separately.

$ 2 \times \frac{2}{3} \times \frac{21}{5} = \frac{2 \times 2 \times 21}{3 \times 5} $

$ = \frac{84}{15} $

Step 7: Simplify the Fraction

Finally, we need to simplify the fraction $ \frac{84}{15} ∗∗.Todothis,wecanfindthegreatestcommondivisor(GCD)ofthenumeratoranddenominator.TheGCDof∗∗**. To do this, we can find the greatest common divisor (GCD) of the numerator and denominator. The GCD of ** 84 ∗∗and∗∗** and ** 15 ∗∗is∗∗** is ** 3 $.

We can divide both the numerator and denominator by the GCD to simplify the fraction.

$ \frac{84}{15} = \frac{84 \div 3}{15 \div 3} $

$ = \frac{28}{5} $

In conclusion, the simplified expression is $ \frac{28}{5} $. This expression is a fraction in its simplest form, and it cannot be simplified further.

The final answer is $ \frac{28}{5} $.
Simplify { 2^{4 / 5} \div \frac{2}{3} \times 4 \frac{1}{5} .} Q&A

In our previous article, we simplified the given mathematical expression: $ 2^{4 / 5} \div \frac{2}{3} \times 4 \frac{1}{5} $. We broke down the expression step by step, using the order of operations and properties of exponents to simplify it. In this article, we will answer some frequently asked questions (FAQs) related to the simplification of this expression.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations is:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions 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.

Q: What is a fractional exponent?

A: A fractional exponent is an exponent that is a fraction. It can be written as $ a^{m/n} = \sqrt[n]{a^m} ∗∗,where∗∗**, where ** a ∗∗isthebaseand∗∗** is the base and ** m/n $ is the exponent.

Q: How do I simplify a radical?

A: To simplify a radical, you need to find the largest perfect square that divides the radicand (the number inside the radical). You can then take the square root of the perfect square and simplify the remaining radicand.

Q: What is the difference between a mixed number and an improper fraction?

A: A mixed number is a number that is written as a combination of a whole number and a fraction. For example, $ 4 \frac{1}{5} ∗∗isamixednumber.Animproperfractionisafractionwherethenumeratorisgreaterthanthedenominator.Forexample,∗∗** is a mixed number. An improper fraction is a fraction where the numerator is greater than the denominator. For example, ** \frac{21}{5} $ is an improper fraction.

Q: How do I multiply fractions?

A: To multiply fractions, you need to multiply the numerators and denominators separately. For example, to multiply $ \frac{2}{3} ∗∗and∗∗** and ** \frac{21}{5} $, you would multiply the numerators and denominators as follows:

$ \frac{2 \times 21}{3 \times 5} = \frac{42}{15} $

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and denominator. You can then divide both the numerator and denominator by the GCD to simplify the fraction.

Q: What is the final answer?

A: The final answer is $ \frac{28}{5} $.

In conclusion, we have answered some frequently asked questions related to the simplification of the given mathematical expression: $ 2^{4 / 5} \div \frac{2}{3} \times 4 \frac{1}{5} $. We hope that this article has been helpful in clarifying any doubts you may have had about the simplification of this expression.

The final answer is $ \frac{28}{5} $.