## Which of the following is equivalent to 60 superscript one-half?

Calculating roots and exponents can be confusing, especially when dealing with large numbers or complex expressions. One common question that arises is, “Which of the following is equivalent to 60 superscript one-half?” Let’s explore the answer and simplify this mathematical expression.

**Understanding the Expression**

The superscript in 60 superscript one-half indicates that we need to raise 60 to the power of one-half. This means we need to find the square root of 60. The square root of a number is the value that, when multiplied by itself, gives us the original number.

**Calculating the Square Root**

Finding the square root of 60 can be done using different methods, such as long division or using a calculator. Using a calculator, we find that the square root of 60 is approximately 7.746.

**Equivalent Expression**

Therefore, the expression that is equivalent to 60 superscript one-half is **7.746**. This means that both expressions have the same value when evaluated.

**Summary**

To find the equivalent of 60 superscript one-half, we need to calculate the square root of 60. Using a calculator, we find that the square root of 60 is approximately 7.746. Therefore, the expression that is equivalent to 60 superscript one-half is 7.746.

## Understanding the Equivalence of 60 Superscript One-Half

### Prelude

In mathematics, exponents are used to denote powers. Understanding the relationship between different exponents is crucial for solving complex equations and simplifying expressions. This article delves into the equivalence of 60 superscript one-half, exploring various approaches to determine its value.

### Powers and Roots

**Power:** Raising a number to a positive exponent indicates repeated multiplication. For instance, 5³ is calculated as 5 × 5 × 5 = 125.

**Root:** Finding the root of a number involves determining the factor that, when multiplied by itself, equals the original number. For example, the square root of 121 is 11, as 11 × 11 = 121.

### Exponents and Radical Form

**Exponents:** In general, a number raised to an exponent of 1/n is equivalent to the nth root of that number. For instance, 60¹/² = √60.

**Radical Form:** The square root of a number can be expressed using the radical symbol √. For example, the square root of 60 can be written as √60.

### Determining the Value of 60 Superscript One-Half

Using the relationship between exponents and radical form, we can determine the value of 60¹/².

**Square Root of 60:**

```
√60 = 60¹/²
```

**Simplifying the Square Root:**

```
√60 = √(4 × 15)
= √4 × √15
= 2√15
```

Therefore, 60¹/² is equivalent to 2√15.

### Alternate Approach: Prime Factorization

**Prime Factorization:** Breaking down a number into its prime factors can help simplify calculations.

**Prime Factors of 60:**

```
60 = 2 × 2 × 3 × 5
```

## Further Understanding of Square Roots

**Perfect Square Roots:** Some numbers have perfect square roots, which are integers. For instance, the square root of 144 is 12 because 12 × 12 = 144.

**Irrational Roots:** Other numbers have irrational square roots, which cannot be expressed as a fraction or decimal. For example, the square root of 2 is an irrational number.

### Practical Applications

**Solving Equations:** Exponents and square roots are used to solve various equations involving powers and roots.

**Geometry:** Understanding square roots is essential for solving problems in geometry, such as finding the area or perimeter of shapes involving right angles.

**Physics:** Square roots appear in formulas used to calculate velocity, acceleration, and other physical quantities.

### Conclusion

In conclusion, 60 superscript one-half is equivalent to 2√15, as determined through the relationship between exponents and radical form. Understanding this equivalence is fundamental for solving problems involving roots and exponents. The concept of square roots has wide-ranging applications in various fields, including mathematics, physics, and geometry.

## Frequently Asked Questions (FAQs)

**1. What is the exponent used for in 60¹/²?**

- The exponent 1/2 indicates that the number is raised to the power of one-half, which is the same as finding its square root.

**2. How do I simplify the square root of 60?**

- Factorize 60 into prime factors (2 × 2 × 3 × 5) and group them into pairs of equal factors, then take the square root of each pair to simplify the expression.

**3. Can all square roots be expressed as integers?**

- No, there are some numbers with irrational square roots that cannot be expressed as fractions or decimals. For example, the square root of 2 is irrational.

**4. What are the practical applications of square roots?**

- Square roots are used in various fields, including problem-solving, geometry (for area and perimeter calculations), and physics (for velocity and acceleration calculations).

**5. How do I calculate the cube root of a number?**

- To find the cube root of a number, use the exponent 1/3. For example, the cube root of 64 is 64¹/³, which is equivalent to 4.

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