What Is The Value Of The Given Expression 8 1 2 ⋅ 8 7 6 8^{\frac{1}{2}} \cdot 8^{\frac{7}{6}} 8 2 1 ​ ⋅ 8 6 7 ​ ? □ \square □

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Introduction

When dealing with exponents, it's essential to understand the rules of exponentiation to simplify complex expressions. In this article, we will explore the value of the given expression $8^{\frac{1}{2}} \cdot 8^{\frac{7}{6}}$ using the properties of exponents.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $a^b$ represents the product of $a$ multiplied by itself $b$ times. In the given expression, we have two exponents: $\frac{1}{2}$ and $\frac{7}{6}$. To simplify the expression, we need to understand the properties of exponents.

Properties of Exponents

There are several properties of exponents that we need to know to simplify the given expression. These properties are:

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

Simplifying the Expression

Using the properties of exponents, we can simplify the given expression as follows:

$$8^{\frac{1}{2}} \cdot 8^{\frac{7}{6}} = 8^{\frac{1}{2} + \frac{7}{6}}$$

Adding the Exponents

To add the exponents, we need to find a common denominator. The least common multiple of 2 and 6 is 6. Therefore, we can rewrite the exponents as follows:

$$8^{\frac{1}{2} + \frac{7}{6}} = 8^{\frac{3}{6} + \frac{7}{6}}$$

Simplifying the Exponents

Now we can add the exponents:

$$8^{\frac{3}{6} + \frac{7}{6}} = 8^{\frac{10}{6}}$$

Simplifying the Fraction

We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

$$8^{\frac{10}{6}} = 8^{\frac{5}{3}}$$

Evaluating the Expression

Now we can evaluate the expression by raising 8 to the power of $\frac{5}{3}$:

$$8^{\frac{5}{3}} = \sqrt[3]{8^5}$$

Simplifying the Radical

We can simplify the radical by evaluating the expression inside the radical:

$$\sqrt[3]{8^5} = \sqrt[3]{32768}$$

Evaluating the Radical

Finally, we can evaluate the radical by finding the cube root of 32768:

$$\sqrt[3]{32768} = 32$$

Conclusion

In conclusion, the value of the given expression $8^{\frac{1}{2}} \cdot 8^{\frac{7}{6}}$ is 32. We simplified the expression using the properties of exponents and evaluated the resulting radical to find the final answer.

Frequently Asked Questions

• Q: What is the value of $8^{\frac{1}{2}}$? A: The value of $8^{\frac{1}{2}}$ is $\sqrt{8}$, which is equal to 2$\sqrt{2}$.
• Q: What is the value of $8^{\frac{7}{6}}$? A: The value of $8^{\frac{7}{6}}$ is $\sqrt[6]{8^7}$, which is equal to $\sqrt[6]{2097152}$.
• Q: How do you simplify the expression $8^{\frac{1}{2}} \cdot 8^{\frac{7}{6}}$? A: To simplify the expression, you can use the properties of exponents to add the exponents and then evaluate the resulting radical.

Final Answer

The final answer is: $\boxed{32}$

Introduction

In our previous article, we explored the value of the given expression $8^{\frac{1}{2}} \cdot 8^{\frac{7}{6}}$. We used the properties of exponents to simplify the expression and evaluate the resulting radical. In this article, we will answer some frequently asked questions related to exponents and simplifying expressions.

Q&A

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

A: A power is the result of raising a number to a certain power, while an exponent is the number that is being raised to a certain power. For example, $a^b$ is a power, while $b$ is an exponent.

Q: How do you simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, you can use the properties of exponents to add or subtract the exponents. For example, $a^b \cdot a^c = a^{b+c}$.

Q: What is the rule for multiplying powers with the same base?

A: When multiplying powers with the same base, you add the exponents. For example, $a^b \cdot a^c = a^{b+c}$.

Q: What is the rule for raising a power to another power?

A: When raising a power to another power, you multiply the exponents. For example, $(a^b)^c = a^{b \cdot c}$.

Q: What is the value of $a^0$?

A: The value of $a^0$ is 1, for any non-zero number $a$.

Q: How do you simplify an expression with a zero exponent?

A: To simplify an expression with a zero exponent, you can use the rule that $a^0 = 1$, for any non-zero number $a$.

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

A: A radical is a mathematical operation that involves finding the root of a number, while an exponent is a number that is being raised to a certain power. For example, $\sqrt[n]{a}$ is a radical, while $a^b$ is an exponent.

Q: How do you simplify an expression with a radical?

A: To simplify an expression with a radical, you can use the properties of radicals to combine or simplify the radical. For example, $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}$.

Q: What is the value of $\sqrt[n]{a}$?

A: The value of $\sqrt[n]{a}$ is the number that, when raised to the power of $n$, equals $a$. For example, $\sqrt[3]{8} = 2$, because $2^3 = 8$.

Q: How do you simplify an expression with multiple radicals?

A: To simplify an expression with multiple radicals, you can use the properties of radicals to combine or simplify the radicals. For example, $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}$.

Conclusion

In conclusion, understanding exponents and simplifying expressions is an essential part of mathematics. By using the properties of exponents and radicals, you can simplify complex expressions and evaluate the resulting values. We hope that this Q&A article has helped you to better understand exponents and simplifying expressions.

Frequently Asked Questions

• Q: What is the value of $8^{\frac{1}{2}}$? A: The value of $8^{\frac{1}{2}}$ is $\sqrt{8}$, which is equal to 2$\sqrt{2}$.
• Q: What is the value of $8^{\frac{7}{6}}$? A: The value of $8^{\frac{7}{6}}$ is $\sqrt[6]{8^7}$, which is equal to $\sqrt[6]{2097152}$.
• Q: How do you simplify the expression $8^{\frac{1}{2}} \cdot 8^{\frac{7}{6}}$? A: To simplify the expression, you can use the properties of exponents to add the exponents and then evaluate the resulting radical.

Final Answer

The final answer is: $\boxed{32}$