Which Of The Following Is Equivalent To $60^{\frac{1}{2}}$?A. $\frac{60}{2}$B. $ 60 \sqrt{60} 60 ​ [/tex]C. $\frac{1}{60^2}$D. $\frac{1}{\sqrt{60}}$

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Introduction

In mathematics, exponents and square roots are fundamental concepts that help us simplify complex expressions and solve equations. Exponents represent repeated multiplication, while square roots represent the inverse operation of squaring a number. In this article, we will explore the concept of equivalent expressions, specifically focusing on the expression $60^{\frac{1}{2}}$ and its equivalent forms.

What is an Exponent?

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 $2^3$, the exponent 3 indicates that the base number 2 is multiplied by itself 3 times, resulting in $2 \times 2 \times 2 = 8$.

What is a Square Root?

A square root is the inverse operation of squaring a number. It represents a number that, when multiplied by itself, gives a specified value. For example, the square root of 16 is 4, because $4 \times 4 = 16$.

Equivalent Expressions

An equivalent expression is an expression that has the same value as another expression. In the context of exponents and square roots, equivalent expressions can be obtained by applying various mathematical operations, such as multiplication, division, addition, and subtraction.

Option A: $\frac{60}{2}$

Option A represents the expression $\frac{60}{2}$. This expression is obtained by dividing 60 by 2, resulting in 30. However, this expression is not equivalent to $60^{\frac{1}{2}}$, as it does not involve any exponentiation or square root operation.

Option B: $\sqrt{60}$

Option B represents the expression $\sqrt{60}$. This expression is obtained by taking the square root of 60, which is a mathematical operation that involves finding a number that, when multiplied by itself, gives 60. However, this expression is not equivalent to $60^{\frac{1}{2}}$, as it does not involve any exponentiation.

Option C: $\frac{1}{60^2}$

Option C represents the expression $\frac{1}{60^2}$. This expression is obtained by taking the reciprocal of $60^2$, which is equivalent to dividing 1 by $60^2$. However, this expression is not equivalent to $60^{\frac{1}{2}}$, as it involves squaring the base number 60, rather than taking its square root.

Option D: $\frac{1}{\sqrt{60}}$

Option D represents the expression $\frac{1}{\sqrt{60}}$. This expression is obtained by taking the reciprocal of the square root of 60, which is equivalent to dividing 1 by the square root of 60. This expression is equivalent to $60^{\frac{1}{2}}$, as it involves taking the reciprocal of the square root of 60, which is the same as taking the square root of 60 and then taking its reciprocal.

Conclusion

In conclusion, the correct answer is Option D: $\frac{1}{\sqrt{60}}$. This expression is equivalent to $60^{\frac{1}{2}}$, as it involves taking the reciprocal of the square root of 60, which is the same as taking the square root of 60 and then taking its reciprocal. Understanding equivalent expressions is crucial in mathematics, as it helps us simplify complex expressions and solve equations.

Real-World Applications

Understanding equivalent expressions has numerous real-world applications. For example, in finance, equivalent expressions can be used to calculate interest rates and investment returns. In engineering, equivalent expressions can be used to design and optimize systems. In science, equivalent expressions can be used to model and analyze complex phenomena.

Tips and Tricks

When working with exponents and square roots, it's essential to remember the following tips and tricks:

  • Exponents represent repeated multiplication, while square roots represent the inverse operation of squaring a number.
  • Equivalent expressions can be obtained by applying various mathematical operations, such as multiplication, division, addition, and subtraction.
  • When working with exponents and square roots, it's essential to simplify expressions by combining like terms and canceling out common factors.

Common Mistakes

When working with exponents and square roots, it's essential to avoid common mistakes, such as:

  • Confusing exponents with square roots.
  • Failing to simplify expressions by combining like terms and canceling out common factors.
  • Not using the correct order of operations (PEMDAS).

Conclusion

Q: What is the difference between an exponent and a square root?

A: An exponent represents repeated multiplication, while a square root represents the inverse operation of squaring a number. For example, in the expression $2^3$, the exponent 3 indicates that the base number 2 is multiplied by itself 3 times, resulting in $2 \times 2 \times 2 = 8$. On the other hand, the square root of 16 is 4, because $4 \times 4 = 16$.

Q: How do I simplify an expression with an exponent?

A: To simplify an expression with an exponent, you can use the following steps:

  1. Identify the base number and the exponent.
  2. Multiply the base number by itself as many times as indicated by the exponent.
  3. Simplify the resulting expression by combining like terms and canceling out common factors.

Q: How do I simplify an expression with a square root?

A: To simplify an expression with a square root, you can use the following steps:

  1. Identify the number inside the square root.
  2. Find a perfect square that is equal to or greater than the number inside the square root.
  3. Take the square root of the perfect square and simplify the resulting expression.

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

A: The order of operations (PEMDAS) is a set of rules that dictates the order in which mathematical operations should be performed. The acronym PEMDAS stands for:

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

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

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

  1. Identify the base numbers and exponents.
  2. Multiply the base numbers together and add the exponents.
  3. Simplify the resulting expression by combining like terms and canceling out common factors.

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

A: A positive exponent indicates that the base number is multiplied by itself as many times as indicated by the exponent. On the other hand, a negative exponent indicates that the base number is divided by itself as many times as indicated by the exponent.

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

A: To evaluate an expression with a negative exponent, you can use the following steps:

  1. Identify the base number and the negative exponent.
  2. Take the reciprocal of the base number and change the sign of the exponent.
  3. Simplify the resulting expression by combining like terms and canceling out common factors.

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

A: A rational exponent is an exponent that can be expressed as a fraction, such as $\frac{1}{2}$ or $\frac{3}{4}$. On the other hand, an irrational exponent is an exponent that cannot be expressed as a fraction, such as $\sqrt{2}$ or $\pi$.

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

A: To evaluate an expression with a rational exponent, you can use the following steps:

  1. Identify the base number and the rational exponent.
  2. Take the base number to the power indicated by the numerator and then take the reciprocal of the result to the power indicated by the denominator.
  3. Simplify the resulting expression by combining like terms and canceling out common factors.

Q: What is the difference between a real number and an imaginary number?

A: A real number is a number that can be expressed as a fraction or a decimal, such as $\frac{1}{2}$ or $3.14$. On the other hand, an imaginary number is a number that cannot be expressed as a fraction or a decimal, such as $i$ or $\sqrt{-1}$.

Q: How do I evaluate an expression with an imaginary number?

A: To evaluate an expression with an imaginary number, you can use the following steps:

  1. Identify the imaginary number and the operation.
  2. Use the properties of imaginary numbers to simplify the expression.
  3. Simplify the resulting expression by combining like terms and canceling out common factors.

Conclusion

In conclusion, understanding exponents and square roots is crucial in mathematics, as it helps us simplify complex expressions and solve equations. By applying various mathematical operations, such as multiplication, division, addition, and subtraction, we can obtain equivalent expressions that have the same value as the original expression. By following the tips and tricks outlined in this article, we can avoid common mistakes and become proficient in working with exponents and square roots.