Use The Properties Of Exponents To Determine The Value Of $a$ For The Equation:$\left(x^{\frac{1}{2}}\right)^5 \sqrt{x^5}=x^a$

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Introduction

In mathematics, exponents are a fundamental concept used to represent repeated multiplication of a number. The properties of exponents are essential in simplifying expressions and solving equations involving exponents. In this article, we will explore how to use the properties of exponents to determine the value of $a$ for the equation $\left(x^{\frac{1}{2}}\right)^5 \sqrt{x^5}=x^a$. We will delve into the world of exponents, explore the rules and properties, and apply them to solve the given equation.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication of a number. For example, $x^3$ can be written as $x \cdot x \cdot x$. The exponent $3$ indicates that the base $x$ is multiplied by itself $3$ times. Exponents can also be used to represent negative numbers, fractions, and decimals.

Properties of Exponents

There are several properties of exponents that are essential in simplifying expressions and solving equations involving exponents. These properties include:

• Product of Powers: When multiplying two powers with the same base, add the exponents. For example, $x^2 \cdot x^3 = x^{2+3} = x^5$.
• Power of a Power: When raising a power to another power, multiply the exponents. For example, $(x^2)^3 = x^{2 \cdot 3} = x^6$.
• Quotient of Powers: When dividing two powers with the same base, subtract the exponents. For example, $\frac{x^2}{x^3} = x^{2-3} = x^{-1}$.
• Zero Exponent: Any non-zero number raised to the power of zero is equal to $1$. For example, $x^0 = 1$.

Simplifying the Equation

Now that we have a good understanding of exponents and their properties, let's simplify the given equation $\left(x^{\frac{1}{2}}\right)^5 \sqrt{x^5}=x^a$. We will start by simplifying the left-hand side of the equation.

Simplifying the Left-Hand Side

The left-hand side of the equation can be simplified using the properties of exponents. We will start by simplifying the expression $\left(x^{\frac{1}{2}}\right)^5$.

$\left(x^{\frac{1}{2}}\right)^5 = x^{\frac{1}{2} \cdot 5} = x^{\frac{5}{2}}$

Next, we will simplify the expression $\sqrt{x^5}$.

$\sqrt{x^5} = x^{\frac{5}{2}}$

Now that we have simplified both expressions, we can rewrite the left-hand side of the equation as:

$\left(x^{\frac{1}{2}}\right)^5 \sqrt{x^5} = x^{\frac{5}{2}} \cdot x^{\frac{5}{2}} = x^{\frac{5}{2} + \frac{5}{2}} = x^5$

Solving for $a$

Now that we have simplified the left-hand side of the equation, we can equate it to the right-hand side and solve for $a$.

$x^5 = x^a$

Since the bases are the same, we can equate the exponents.

$5 = a$

Therefore, the value of $a$ is $5$.

Conclusion

In this article, we used the properties of exponents to determine the value of $a$ for the equation $\left(x^{\frac{1}{2}}\right)^5 \sqrt{x^5}=x^a$. We simplified the left-hand side of the equation using the properties of exponents and then equated it to the right-hand side to solve for $a$. The value of $a$ is $5$. This problem demonstrates the importance of understanding the properties of exponents and how they can be used to simplify expressions and solve equations involving exponents.

Frequently Asked Questions

• What is the value of $a$ for the equation $\left(x^{\frac{1}{2}}\right)^5 \sqrt{x^5}=x^a$?
  • The value of $a$ is $5$.
• How do you simplify the expression $\left(x^{\frac{1}{2}}\right)^5$?
  • You can simplify the expression $\left(x^{\frac{1}{2}}\right)^5$ by using the property of exponents: $\left(x^{\frac{1}{2}}\right)^5 = x^{\frac{1}{2} \cdot 5} = x^{\frac{5}{2}}$.
• How do you simplify the expression $\sqrt{x^5}$?
  • You can simplify the expression $\sqrt{x^5}$ by using the property of exponents: $\sqrt{x^5} = x^{\frac{5}{2}}$.

References

• "Algebra and Trigonometry" by Michael Sullivan
• "College Algebra" by James Stewart
• "Precalculus" by Michael Sullivan

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources.

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Introduction

In our previous article, we used the properties of exponents to determine the value of $a$ for the equation $\left(x^{\frac{1}{2}}\right)^5 \sqrt{x^5}=x^a$. We simplified the left-hand side of the equation using the properties of exponents and then equated it to the right-hand side to solve for $a$. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q: What is the value of $a$ for the equation $\left(x^{\frac{1}{2}}\right)^5 \sqrt{x^5}=x^a$?

A: The value of $a$ is $5$.

Q: How do you simplify the expression $\left(x^{\frac{1}{2}}\right)^5$?

A: You can simplify the expression $\left(x^{\frac{1}{2}}\right)^5$ by using the property of exponents: $\left(x^{\frac{1}{2}}\right)^5 = x^{\frac{1}{2} \cdot 5} = x^{\frac{5}{2}}$.

Q: How do you simplify the expression $\sqrt{x^5}$?

A: You can simplify the expression $\sqrt{x^5}$ by using the property of exponents: $\sqrt{x^5} = x^{\frac{5}{2}}$.

Q: What is the product of powers property of exponents?

A: The product of powers property of exponents states that when multiplying two powers with the same base, add the exponents. For example, $x^2 \cdot x^3 = x^{2+3} = x^5$.

Q: What is the power of a power property of exponents?

A: The power of a power property of exponents states that when raising a power to another power, multiply the exponents. For example, $(x^2)^3 = x^{2 \cdot 3} = x^6$.

Q: What is the quotient of powers property of exponents?

A: The quotient of powers property of exponents states that when dividing two powers with the same base, subtract the exponents. For example, $\frac{x^2}{x^3} = x^{2-3} = x^{-1}$.

Q: What is the zero exponent property of exponents?

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

Tips and Tricks

• Use the properties of exponents to simplify expressions: When simplifying expressions involving exponents, use the properties of exponents to combine like terms and simplify the expression.
• Use the properties of exponents to solve equations: When solving equations involving exponents, use the properties of exponents to isolate the variable and solve for the value of the variable.
• Practice, practice, practice: The more you practice using the properties of exponents, the more comfortable you will become with applying them to simplify expressions and solve equations.

Conclusion

In this article, we answered some frequently asked questions related to using the properties of exponents to determine the value of $a$ for the equation $\left(x^{\frac{1}{2}}\right)^5 \sqrt{x^5}=x^a$. We also provided some tips and tricks for using the properties of exponents to simplify expressions and solve equations. By following these tips and tricks, you will become more comfortable using the properties of exponents and be able to apply them to a wide range of problems.

Frequently Asked Questions

• What is the value of $a$ for the equation $\left(x^{\frac{1}{2}}\right)^5 \sqrt{x^5}=x^a$?
  • The value of $a$ is $5$.
• How do you simplify the expression $\left(x^{\frac{1}{2}}\right)^5$?
  • You can simplify the expression $\left(x^{\frac{1}{2}}\right)^5$ by using the property of exponents: $\left(x^{\frac{1}{2}}\right)^5 = x^{\frac{1}{2} \cdot 5} = x^{\frac{5}{2}}$.
• How do you simplify the expression $\sqrt{x^5}$?
  • You can simplify the expression $\sqrt{x^5}$ by using the property of exponents: $\sqrt{x^5} = x^{\frac{5}{2}}$.

References

• "Algebra and Trigonometry" by Michael Sullivan
• "College Algebra" by James Stewart
• "Precalculus" by Michael Sullivan

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources.