Simplify: 12 5 ⋅ 8 5 \sqrt[5]{12} \cdot \sqrt[5]{8} 5 12 ​ ⋅ 5 8 ​

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Introduction

When dealing with exponents and roots, it's often helpful to simplify expressions before performing operations. In this case, we're given the expression $\sqrt[5]{12} \cdot \sqrt[5]{8}$, and we want to simplify it. To do this, we'll use the properties of exponents and roots to rewrite the expression in a more manageable form.

Understanding the Properties of Exponents and Roots

Before we dive into simplifying the expression, let's review the properties of exponents and roots that we'll need to use. The property we'll be using most is the product of powers property, which states that when we multiply two numbers with the same base, we can add their exponents. In other words, if we have $a^m \cdot a^n$, we can rewrite it as $a^{m+n}$.

We'll also be using the property of roots, which states that $\sqrt[n]{a} = a^{\frac{1}{n}}$. This means that we can rewrite a root as an exponent.

Simplifying the Expression

Now that we've reviewed the properties of exponents and roots, let's simplify the expression $\sqrt[5]{12} \cdot \sqrt[5]{8}$. To do this, we'll use the property of roots to rewrite each root as an exponent.

$\sqrt[5]{12} \cdot \sqrt[5]{8} = (12)^{\frac{1}{5}} \cdot (8)^{\frac{1}{5}}$

Using the Product of Powers Property

Now that we've rewritten each root as an exponent, we can use the product of powers property to simplify the expression. This property states that when we multiply two numbers with the same base, we can add their exponents.

$(12)^{\frac{1}{5}} \cdot (8)^{\frac{1}{5}} = (12 \cdot 8)^{\frac{1}{5}}$

Simplifying the Expression Further

Now that we've used the product of powers property, we can simplify the expression further. To do this, we'll multiply the numbers inside the parentheses.

$(12 \cdot 8)^{\frac{1}{5}} = (96)^{\frac{1}{5}}$

Using the Property of Roots Again

Now that we've simplified the expression, we can use the property of roots again to rewrite the root as an exponent.

$(96)^{\frac{1}{5}} = \sqrt[5]{96}$

Conclusion

In this article, we simplified the expression $\sqrt[5]{12} \cdot \sqrt[5]{8}$ using the properties of exponents and roots. We used the product of powers property to rewrite the expression as a single exponent, and then used the property of roots to rewrite the exponent as a root. The final simplified expression is $\sqrt[5]{96}$.

Final Answer

The final answer is $\boxed{\sqrt[5]{96}}$.

Additional Examples

Here are a few additional examples of simplifying expressions using the properties of exponents and roots.

Example 1

Simplify the expression $\sqrt[3]{27} \cdot \sqrt[3]{8}$.

$\sqrt[3]{27} \cdot \sqrt[3]{8} = (27)^{\frac{1}{3}} \cdot (8)^{\frac{1}{3}}$

Using the product of powers property, we get:

$(27)^{\frac{1}{3}} \cdot (8)^{\frac{1}{3}} = (27 \cdot 8)^{\frac{1}{3}}$

Simplifying the expression further, we get:

$(27 \cdot 8)^{\frac{1}{3}} = (216)^{\frac{1}{3}}$

Using the property of roots again, we get:

$(216)^{\frac{1}{3}} = \sqrt[3]{216}$

The final answer is $\boxed{\sqrt[3]{216}}$.

Example 2

Simplify the expression $\sqrt[4]{16} \cdot \sqrt[4]{9}$.

$\sqrt[4]{16} \cdot \sqrt[4]{9} = (16)^{\frac{1}{4}} \cdot (9)^{\frac{1}{4}}$

Using the product of powers property, we get:

$(16)^{\frac{1}{4}} \cdot (9)^{\frac{1}{4}} = (16 \cdot 9)^{\frac{1}{4}}$

Simplifying the expression further, we get:

$(16 \cdot 9)^{\frac{1}{4}} = (144)^{\frac{1}{4}}$

Using the property of roots again, we get:

$(144)^{\frac{1}{4}} = \sqrt[4]{144}$

The final answer is $\boxed{\sqrt[4]{144}}$.

Conclusion

In this article, we simplified two expressions using the properties of exponents and roots. We used the product of powers property to rewrite the expressions as single exponents, and then used the property of roots to rewrite the exponents as roots. The final simplified expressions were $\sqrt[3]{216}$ and $\sqrt[4]{144}$.

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Introduction

In our previous article, we simplified the expression $\sqrt[5]{12} \cdot \sqrt[5]{8}$ using the properties of exponents and roots. We used the product of powers property to rewrite the expression as a single exponent, and then used the property of roots to rewrite the exponent as a root. The final simplified expression was $\sqrt[5]{96}$.

In this article, we'll answer some common questions related to simplifying expressions using the properties of exponents and roots.

Q&A

Q: What is the product of powers property?

A: The product of powers property states that when we multiply two numbers with the same base, we can add their exponents. In other words, if we have $a^m \cdot a^n$, we can rewrite it as $a^{m+n}$.

Q: How do I use the product of powers property to simplify an expression?

A: To use the product of powers property, we need to identify the base and the exponents in the expression. We can then add the exponents and rewrite the expression as a single exponent.

Q: What is the property of roots?

A: The property of roots states that $\sqrt[n]{a} = a^{\frac{1}{n}}$. This means that we can rewrite a root as an exponent.

Q: How do I use the property of roots to simplify an expression?

A: To use the property of roots, we need to rewrite the root as an exponent. We can then use the product of powers property to simplify the expression further.

Q: Can I use the product of powers property and the property of roots together to simplify an expression?

A: Yes, we can use both properties together to simplify an expression. We can use the product of powers property to rewrite the expression as a single exponent, and then use the property of roots to rewrite the exponent as a root.

Q: What are some common mistakes to avoid when simplifying expressions using the properties of exponents and roots?

A: Some common mistakes to avoid include:

• Not identifying the base and the exponents in the expression
• Not adding the exponents correctly when using the product of powers property
• Not rewriting the root as an exponent when using the property of roots
• Not simplifying the expression further after using the product of powers property and the property of roots

Example Questions

Q: Simplify the expression $\sqrt[3]{27} \cdot \sqrt[3]{8}$.

A: To simplify this expression, we can use the product of powers property and the property of roots. We can rewrite the expression as:

$\sqrt[3]{27} \cdot \sqrt[3]{8} = (27)^{\frac{1}{3}} \cdot (8)^{\frac{1}{3}}$

Using the product of powers property, we get:

$(27)^{\frac{1}{3}} \cdot (8)^{\frac{1}{3}} = (27 \cdot 8)^{\frac{1}{3}}$

Simplifying the expression further, we get:

$(27 \cdot 8)^{\frac{1}{3}} = (216)^{\frac{1}{3}}$

Using the property of roots again, we get:

$(216)^{\frac{1}{3}} = \sqrt[3]{216}$

The final answer is $\boxed{\sqrt[3]{216}}$.

Q: Simplify the expression $\sqrt[4]{16} \cdot \sqrt[4]{9}$.

A: To simplify this expression, we can use the product of powers property and the property of roots. We can rewrite the expression as:

$\sqrt[4]{16} \cdot \sqrt[4]{9} = (16)^{\frac{1}{4}} \cdot (9)^{\frac{1}{4}}$

Using the product of powers property, we get:

$(16)^{\frac{1}{4}} \cdot (9)^{\frac{1}{4}} = (16 \cdot 9)^{\frac{1}{4}}$

Simplifying the expression further, we get:

$(16 \cdot 9)^{\frac{1}{4}} = (144)^{\frac{1}{4}}$

Using the property of roots again, we get:

$(144)^{\frac{1}{4}} = \sqrt[4]{144}$

The final answer is $\boxed{\sqrt[4]{144}}$.

Conclusion

In this article, we answered some common questions related to simplifying expressions using the properties of exponents and roots. We used the product of powers property and the property of roots to simplify two expressions, and we identified some common mistakes to avoid when simplifying expressions. We hope this article has been helpful in understanding how to simplify expressions using the properties of exponents and roots.