Simplify The Following Expression:${ \left[\left{\left(\frac{5}{9}\right) {-\frac{1}{4}}\right} {\frac{8}{3}}\right]^{-\frac{3}{2}} }$

by ADMIN 136 views

=====================================================

Introduction

In this article, we will simplify a given mathematical expression involving exponents and fractions. The expression is [{(59)−14}83]−32\left[\left\{\left(\frac{5}{9}\right)^{-\frac{1}{4}}\right\}^{\frac{8}{3}}\right]^{-\frac{3}{2}}. We will break down the expression step by step and simplify it using the rules of exponents.

Understanding the Expression

The given expression is a complex fraction involving exponents and parentheses. To simplify it, we need to follow the order of operations (PEMDAS):

  1. Evaluate the expressions inside the parentheses.
  2. Simplify the exponents.
  3. Multiply and divide the fractions.

Step 1: Evaluate the Expression Inside the Parentheses

The expression inside the parentheses is (59)−14\left(\frac{5}{9}\right)^{-\frac{1}{4}}. To evaluate this expression, we need to apply the rule of negative exponents:

(59)−14=914514\left(\frac{5}{9}\right)^{-\frac{1}{4}} = \frac{9^{\frac{1}{4}}}{5^{\frac{1}{4}}}

Step 2: Simplify the Exponents

Now, we need to simplify the exponents:

914514=9454\frac{9^{\frac{1}{4}}}{5^{\frac{1}{4}}} = \frac{\sqrt[4]{9}}{\sqrt[4]{5}}

Step 3: Raise the Expression to the Power of 83\frac{8}{3}

Next, we need to raise the expression to the power of 83\frac{8}{3}:

(9454)83=(94)83(54)83\left(\frac{\sqrt[4]{9}}{\sqrt[4]{5}}\right)^{\frac{8}{3}} = \frac{(\sqrt[4]{9})^{\frac{8}{3}}}{(\sqrt[4]{5})^{\frac{8}{3}}}

Step 4: Simplify the Exponents Again

Now, we need to simplify the exponents again:

(94)83(54)83=923523\frac{(\sqrt[4]{9})^{\frac{8}{3}}}{(\sqrt[4]{5})^{\frac{8}{3}}} = \frac{9^{\frac{2}{3}}}{5^{\frac{2}{3}}}

Step 5: Raise the Expression to the Power of −32-\frac{3}{2}

Finally, we need to raise the expression to the power of −32-\frac{3}{2}:

(923523)−32=(523)32(923)32\left(\frac{9^{\frac{2}{3}}}{5^{\frac{2}{3}}}\right)^{-\frac{3}{2}} = \frac{(5^{\frac{2}{3}})^{\frac{3}{2}}}{(9^{\frac{2}{3}})^{\frac{3}{2}}}

Step 6: Simplify the Final Expression

Now, we can simplify the final expression:

(523)32(923)32=5191\frac{(5^{\frac{2}{3}})^{\frac{3}{2}}}{(9^{\frac{2}{3}})^{\frac{3}{2}}} = \frac{5^1}{9^1}

Conclusion

In this article, we simplified a given mathematical expression involving exponents and fractions. We followed the order of operations (PEMDAS) and applied the rules of exponents to simplify the expression. The final simplified expression is 59\frac{5}{9}.

Frequently Asked Questions

Q: What is the order of operations (PEMDAS)?

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

  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 the rule of negative exponents?

A: The rule of negative exponents states that for any non-zero number aa and any integer nn, a−n=1ana^{-n} = \frac{1}{a^n}.

Q: How do I simplify an expression involving exponents?

A: To simplify an expression involving exponents, follow the order of operations (PEMDAS) and apply the rules of exponents. First, evaluate any expressions inside parentheses. Next, simplify any exponential expressions. Finally, multiply and divide any fractions.

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

====================================================================

Introduction

In our previous article, we simplified a given mathematical expression involving exponents and fractions. In this article, we will answer some frequently asked questions about simplifying mathematical expressions.

Q&A

Q: What is the order of operations (PEMDAS)?

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

  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 the rule of negative exponents?

A: The rule of negative exponents states that for any non-zero number aa and any integer nn, a−n=1ana^{-n} = \frac{1}{a^n}.

Q: How do I simplify an expression involving exponents?

A: To simplify an expression involving exponents, follow the order of operations (PEMDAS) and apply the rules of exponents. First, evaluate any expressions inside parentheses. Next, simplify any exponential expressions. Finally, multiply and divide any fractions.

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

A: A fraction is a way of expressing a part of a whole as a ratio of two numbers. For example, 12\frac{1}{2} is a fraction. A decimal is a way of expressing a number as a sum of powers of 10. For example, 0.50.5 is a decimal.

Q: How do I convert a fraction to a decimal?

A: To convert a fraction to a decimal, divide the numerator by the denominator. For example, to convert 12\frac{1}{2} to a decimal, divide 1 by 2, which gives 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 68\frac{6}{8}, find the GCD of 6 and 8, which is 2. Then, divide both numbers by 2, which gives 34\frac{3}{4}.

Q: What is the difference between a rational number and an irrational number?

A: A rational number is a number that can be expressed as a ratio of two integers. For example, 12\frac{1}{2} is a rational number. An irrational number is a number that cannot be expressed as a ratio of two integers. For example, 2\sqrt{2} is an irrational number.

Q: How do I determine if a number is rational or irrational?

A: To determine if a number is rational or irrational, try to express it as a ratio of two integers. If you can, then the number is rational. If you cannot, then the number is irrational.

Conclusion

In this article, we answered some frequently asked questions about simplifying mathematical expressions. We covered topics such as the order of operations (PEMDAS), the rule of negative exponents, and the difference between rational and irrational numbers.

Frequently Asked Questions

Q: What is the difference between a variable and a constant?

A: A variable is a symbol that represents a value that can change. For example, xx is a variable. A constant is a value that does not change. For example, 5 is a constant.

Q: How do I solve an equation?

A: To solve an equation, isolate the variable on one side of the equation. For example, to solve the equation x+2=5x + 2 = 5, subtract 2 from both sides, which gives x=3x = 3.

Q: What is the difference between a function and an equation?

A: A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). For example, f(x)=x2f(x) = x^2 is a function. An equation is a statement that two expressions are equal. For example, x+2=5x + 2 = 5 is an equation.

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton