Express The Following In Radical Form:4. 140 1 / 8 140^{1/8} 14 0 1/8 (Note: Refer To Example 1 For Guidance.)

Introduction

Radical form is a way of expressing numbers that involve roots, such as square roots, cube roots, and higher-order roots. In this article, we will focus on expressing the number $140^{1/8}$ in radical form. This requires a good understanding of exponent rules and radical notation.

Understanding Exponent Rules

Before we dive into expressing $140^{1/8}$ in radical form, let's review some exponent rules that will be useful in this process.

• Product of Powers Rule: When multiplying two numbers with the same base, we add their exponents. For example, $a^m \cdot a^n = a^{m+n}$.
• Power of a Power Rule: When raising a power to another power, we multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.
• Zero Exponent Rule: Any non-zero number raised to the power of zero is equal to 1. For example, $a^0 = 1$.

Expressing $140^{1/8}$ in Radical Form

Now that we have reviewed the necessary exponent rules, let's focus on expressing $140^{1/8}$ in radical form.

To express $140^{1/8}$ in radical form, we need to find the prime factorization of 140.

Prime Factorization of 140

The prime factorization of 140 is:

$$140 = 2 \cdot 2 \cdot 5 \cdot 7$$

Expressing $140^{1/8}$ in Radical Form

Now that we have the prime factorization of 140, we can express $140^{1/8}$ in radical form.

$$140^{1/8} = (2 \cdot 2 \cdot 5 \cdot 7)^{1/8}$$

Using the product of powers rule, we can rewrite this expression as:

$$140^{1/8} = 2^{1/8} \cdot 2^{1/8} \cdot 5^{1/8} \cdot 7^{1/8}$$

Simplifying the Expression

Now that we have expressed $140^{1/8}$ in radical form, we can simplify the expression by combining the like terms.

$$140^{1/8} = 2^{1/8} \cdot 2^{1/8} \cdot 5^{1/8} \cdot 7^{1/8}$$

Using the product of powers rule, we can rewrite this expression as:

$$140^{1/8} = 2^{2/8} \cdot 5^{1/8} \cdot 7^{1/8}$$

Final Answer

The final answer is:

$$140^{1/8} = 2^{1/4} \cdot 5^{1/8} \cdot 7^{1/8}$$

Conclusion

Expressing numbers in radical form requires a good understanding of exponent rules and radical notation. In this article, we have focused on expressing the number $140^{1/8}$ in radical form. We have reviewed the necessary exponent rules, found the prime factorization of 140, and expressed $140^{1/8}$ in radical form. Finally, we have simplified the expression by combining the like terms.

Example 1: Expressing $216^{1/3}$ in Radical Form

To express $216^{1/3}$ in radical form, we need to find the prime factorization of 216.

The prime factorization of 216 is:

$$216 = 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3$$

Using the product of powers rule, we can rewrite this expression as:

$$216^{1/3} = 2^{1/3} \cdot 2^{1/3} \cdot 2^{1/3} \cdot 3^{1/3} \cdot 3^{1/3} \cdot 3^{1/3}$$

Using the product of powers rule, we can rewrite this expression as:

$$216^{1/3} = 2^{3/3} \cdot 3^{3/3}$$

The final answer is:

$$216^{1/3} = 2 \cdot 3$$

Example 2: Expressing $243^{1/5}$ in Radical Form

To express $243^{1/5}$ in radical form, we need to find the prime factorization of 243.

The prime factorization of 243 is:

$$243 = 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3$$

Using the product of powers rule, we can rewrite this expression as:

$$243^{1/5} = 3^{1/5} \cdot 3^{1/5} \cdot 3^{1/5} \cdot 3^{1/5} \cdot 3^{1/5}$$

Using the product of powers rule, we can rewrite this expression as:

$$243^{1/5} = 3^{5/5}$$

The final answer is:

243^{1/5} = 3$<br/> **Q&A: Expressing Numbers in Radical Form** =============================================

Q: What is radical form?

A: Radical form is a way of expressing numbers that involve roots, such as square roots, cube roots, and higher-order roots.

Q: Why do we need to express numbers in radical form?

A: Expressing numbers in radical form is useful when we need to simplify expressions that involve roots. It can also help us to identify the prime factorization of a number.

Q: How do I express a number in radical form?

A: To express a number in radical form, we need to find the prime factorization of the number. We can then use the product of powers rule to rewrite the expression in radical form.

Q: What is the product of powers rule?

A: The product of powers rule states that when multiplying two numbers with the same base, we add their exponents. For example, $a^m \cdot a^n = a^{m+n}$.

Q: How do I use the product of powers rule to express a number in radical form?

A: To use the product of powers rule to express a number in radical form, we need to find the prime factorization of the number. We can then rewrite the expression using the product of powers rule.

Q: What is the zero exponent rule?

A: The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1. For example, $a^0 = 1$.

Q: How do I use the zero exponent rule to simplify an expression in radical form?

A: To use the zero exponent rule to simplify an expression in radical form, we need to identify any terms that have a zero exponent. We can then simplify the expression by removing these terms.

Q: Can I use the product of powers rule to simplify an expression in radical form?

A: Yes, you can use the product of powers rule to simplify an expression in radical form. This rule states that when multiplying two numbers with the same base, we add their exponents.

Q: How do I use the product of powers rule to simplify an expression in radical form?

A: To use the product of powers rule to simplify an expression in radical form, we need to identify any terms that have the same base. We can then add their exponents and simplify the expression.

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

A: A radical is a symbol that represents a root, such as a square root or a cube root. An exponent, on the other hand, is a number that is raised to a power.

Q: Can I use a calculator to express a number in radical form?

A: Yes, you can use a calculator to express a number in radical form. However, it's often more useful to express numbers in radical form by hand, as this can help you to understand the underlying math.

Q: How do I know when to use radical form?

A: You should use radical form when you need to simplify expressions that involve roots. This can help you to identify the prime factorization of a number and simplify the expression.

Q: Can I use radical form to solve equations?

A: Yes, you can use radical form to solve equations. This can help you to simplify the equation and identify the solution.

Q: How do I use radical form to solve equations?

A: To use radical form to solve equations, you need to simplify the equation and identify the solution. This can involve using the product of powers rule and the zero exponent rule to simplify the expression.

Q: What are some common mistakes to avoid when using radical form?

A: Some common mistakes to avoid when using radical form include:

• Forgetting to simplify the expression
• Not using the product of powers rule and the zero exponent rule
• Not identifying the prime factorization of the number
• Not using the correct notation for radical form

Q: How can I practice using radical form?

A: You can practice using radical form by working through examples and exercises. This can help you to build your skills and confidence when using radical form.

Q: What are some real-world applications of radical form?

A: Radical form has many real-world applications, including:

• Simplifying expressions that involve roots
• Identifying the prime factorization of a number
• Solving equations that involve roots
• Working with mathematical models that involve roots

Q: Can I use radical form to solve problems in other areas of math?

A: Yes, you can use radical form to solve problems in other areas of math, including algebra, geometry, and trigonometry. This can help you to simplify expressions and identify solutions.