Simplify The Expression:${ \left{ \begin{array}{l} \left.(\sqrt{2})^3 \times (\sqrt{2}) {-5}\right} 6 \ \left.(\sqrt{2}) {3+5}\right}_6 6 - 2 \end{array} \right. }$

Introduction

In this article, we will simplify the given expression using the properties of exponents and radicals. The expression involves the use of square roots and exponents, and we will break it down into manageable steps to arrive at the final simplified form.

Understanding the Expression

The given expression is:

$$\left\{ \begin{array}{l} \left.(\sqrt{2})^3 \times (\sqrt{2})^{-5}\right\}^6 \\ \left.(\sqrt{2})^{3+5}\right\}_6^6 - 2 \end{array} \right.$$

This expression involves the use of square roots and exponents. We will start by simplifying the first part of the expression, which involves the product of two terms with exponents.

Simplifying the First Part

The first part of the expression is:

$$(\sqrt{2})^3 \times (\sqrt{2})^{-5}$$

Using the property of exponents that states $a^m \times a^n = a^{m+n}$, we can simplify this expression as follows:

$$(\sqrt{2})^3 \times (\sqrt{2})^{-5} = (\sqrt{2})^{3-5} = (\sqrt{2})^{-2}$$

Now, we can simplify this expression further by using the property of exponents that states $a^{-n} = \frac{1}{a^n}$. This gives us:

$$(\sqrt{2})^{-2} = \frac{1}{(\sqrt{2})^2} = \frac{1}{2}$$

Raising to the Power of 6

The next step is to raise this simplified expression to the power of 6:

$$\left(\frac{1}{2}\right)^6$$

Using the property of exponents that states $(a^m)^n = a^{mn}$, we can simplify this expression as follows:

$$\left(\frac{1}{2}\right)^6 = \frac{1}{2^6} = \frac{1}{64}$$

Simplifying the Second Part

The second part of the expression is:

$$(\sqrt{2})^{3+5}$$

Using the property of exponents that states $a^m \times a^n = a^{m+n}$, we can simplify this expression as follows:

$$(\sqrt{2})^{3+5} = (\sqrt{2})^8$$

Now, we can simplify this expression further by using the property of exponents that states $(a^m)^n = a^{mn}$. This gives us:

$$(\sqrt{2})^8 = \left((\sqrt{2})^2\right)^4 = 2^4 = 16$$

Raising to the Power of 6

The next step is to raise this simplified expression to the power of 6:

$$16^6$$

Using the property of exponents that states $(a^m)^n = a^{mn}$, we can simplify this expression as follows:

$$16^6 = (2^4)^6 = 2^{24}$$

Subtracting 2

The final step is to subtract 2 from the result:

$$2^{24} - 2$$

Using the property of exponents that states $a^m - a^n = a^n(a^{m-n} - 1)$, we can simplify this expression as follows:

$$2^{24} - 2 = 2(2^{23} - 1)$$

Conclusion

In this article, we simplified the given expression using the properties of exponents and radicals. We broke down the expression into manageable steps and arrived at the final simplified form. The final simplified form is:

$$2(2^{23} - 1)$$

Q&A: Simplifying the Expression

Q: What is the final simplified form of the given expression? A: The final simplified form of the given expression is $2(2^{23} - 1)$.

Q: How do I simplify the expression $(\sqrt{2})^3 \times (\sqrt{2})^{-5}$? A: To simplify this expression, we can use the property of exponents that states $a^m \times a^n = a^{m+n}$. This gives us $(\sqrt{2})^3 \times (\sqrt{2})^{-5} = (\sqrt{2})^{3-5} = (\sqrt{2})^{-2}$. We can then simplify this expression further by using the property of exponents that states $a^{-n} = \frac{1}{a^n}$. This gives us $(\sqrt{2})^{-2} = \frac{1}{(\sqrt{2})^2} = \frac{1}{2}$.

Q: How do I raise the simplified expression $\frac{1}{2}$ to the power of 6? A: To raise the simplified expression $\frac{1}{2}$ to the power of 6, we can use the property of exponents that states $(a^m)^n = a^{mn}$. This gives us $\left(\frac{1}{2}\right)^6 = \frac{1}{2^6} = \frac{1}{64}$.

Q: How do I simplify the expression $(\sqrt{2})^{3+5}$? A: To simplify this expression, we can use the property of exponents that states $a^m \times a^n = a^{m+n}$. This gives us $(\sqrt{2})^{3+5} = (\sqrt{2})^8$. We can then simplify this expression further by using the property of exponents that states $(a^m)^n = a^{mn}$. This gives us $(\sqrt{2})^8 = \left((\sqrt{2})^2\right)^4 = 2^4 = 16$.

Q: How do I raise the simplified expression 16 to the power of 6? A: To raise the simplified expression 16 to the power of 6, we can use the property of exponents that states $(a^m)^n = a^{mn}$. This gives us $16^6 = (2^4)^6 = 2^{24}$.

Q: How do I subtract 2 from the result $2^{24}$? A: To subtract 2 from the result $2^{24}$, we can use the property of exponents that states $a^m - a^n = a^n(a^{m-n} - 1)$. This gives us $2^{24} - 2 = 2(2^{23} - 1)$.

Q: What are some common properties of exponents that I can use to simplify expressions? A: Some common properties of exponents that you can use to simplify expressions include:

• $a^m \times a^n = a^{m+n}$
• $(a^m)^n = a^{mn}$
• $a^{-n} = \frac{1}{a^n}$
• $a^m - a^n = a^n(a^{m-n} - 1)$

Q: How can I apply these properties of exponents to simplify expressions? A: To apply these properties of exponents to simplify expressions, you can follow these steps:

1. Identify the properties of exponents that can be used to simplify the expression.
2. Apply the properties of exponents to simplify the expression.
3. Simplify the expression further by using other properties of exponents.
4. Check the final simplified form to ensure that it is correct.

Conclusion

In this article, we simplified the given expression using the properties of exponents and radicals. We broke down the expression into manageable steps and arrived at the final simplified form. We also answered some common questions about simplifying expressions using the properties of exponents.