Evaluate The Left-hand Side To Find The Value Of $a$ In The Equation In Simplest Form.$\frac{x^{\frac{5}{4}}}{x^{\frac{1}{6}}} = X^a$

Feb 28, 2025 by ADMIN 136 views

Simplifying Exponents: A Step-by-Step Guide to Evaluating the Left-Hand Side of the Equation

In mathematics, exponents play a crucial role in representing large numbers in a compact form. When dealing with equations involving exponents, it's essential to simplify them to their simplest form to make it easier to solve for the unknown variable. In this article, we will focus on evaluating the left-hand side of the equation $\frac{x^{\frac{5}{4}}}{x^{\frac{1}{6}}} = x^a$ to find the value of $a$ in its simplest form.

Before we dive into simplifying the equation, let's take a moment to understand what exponents represent. An exponent is a small number that is raised to a power, indicating how many times a base number is multiplied by itself. For example, $x^2$ means $x$ multiplied by itself, or $x \times x$ . Similarly, $x^3$ means $x$ multiplied by itself three times, or $x \times x \times x$ .

Simplifying the Left-Hand Side of the Equation

Now that we have a basic understanding of exponents, let's simplify the left-hand side of the equation $\frac{x^{\frac{5}{4}}}{x^{\frac{1}{6}}}$ . To do this, we need to apply the rule of dividing exponents with the same base, which states that when dividing two exponents with the same base, we subtract the exponents.

$\frac{x^{\frac{5}{4}}}{x^{\frac{1}{6}}} = x^{\frac{5}{4} - \frac{1}{6}}$

To simplify the expression, we need to find a common denominator for the fractions. The least common multiple (LCM) of 4 and 6 is 12. So, we can rewrite the fractions with a common denominator of 12.

$\frac{5}{4} = \frac{5 \times 3}{4 \times 3} = \frac{15}{12}$

$\frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}$

Now, we can subtract the exponents.

$x^{\frac{5}{4} - \frac{1}{6}} = x^{\frac{15}{12} - \frac{2}{12}} = x^{\frac{13}{12}}$

Evaluating the Right-Hand Side of the Equation

Now that we have simplified the left-hand side of the equation, let's evaluate the right-hand side. The right-hand side of the equation is $x^a$ , where $a$ is the unknown variable we are trying to solve for.

Equating the Left-Hand and Right-Hand Sides

Now that we have simplified the left-hand side of the equation and evaluated the right-hand side, we can equate the two expressions.

$x^{\frac{13}{12}} = x^a$

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

$\frac{13}{12} = a$

In conclusion, we have successfully simplified the left-hand side of the equation $\frac{x^{\frac{5}{4}}}{x^{\frac{1}{6}}} = x^a$ to find the value of $a$ in its simplest form. By applying the rule of dividing exponents with the same base and simplifying the expression, we were able to equate the left-hand and right-hand sides of the equation and solve for $a$ . The value of $a$ is $\frac{13}{12}$ .

The final answer is $\boxed{\frac{13}{12}}$ .
Simplifying Exponents: A Step-by-Step Guide to Evaluating the Left-Hand Side of the Equation (Q&A)

In our previous article, we explored the concept of simplifying exponents and applied it to the equation $\frac{x^{\frac{5}{4}}}{x^{\frac{1}{6}}} = x^a$ . We successfully simplified the left-hand side of the equation to find the value of $a$ in its simplest form. In this article, we will address some common questions and concerns related to simplifying exponents.

Q: What is the rule for dividing exponents with the same base?

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

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you need to apply the rules of exponents. These rules include:

Multiplying exponents with the same base: $x^a \times x^b = x^{a+b}$
Dividing exponents with the same base: $\frac{x^a}{x^b} = x^{a-b}$
Raising an exponent to a power: $(x^a)^b = x^{ab}$

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent indicates that the base is being multiplied by itself a certain number of times. For example, $x^2$ means $x$ multiplied by itself two times. A negative exponent indicates that the base is being divided by itself a certain number of times. For example, $x^{-2}$ means $x$ divided by itself two times.

Q: How do I evaluate an expression with a negative exponent?

A: To evaluate an expression with a negative exponent, you need to rewrite the expression with a positive exponent. For example, $x^{-2}$ can be rewritten as $\frac{1}{x^2}$ .

Q: What is the order of operations when simplifying exponents?

A: The order of operations when simplifying exponents is:

Parentheses: Evaluate any expressions inside parentheses first.
Exponents: Evaluate any exponents next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: Can I simplify an expression with multiple exponents?

A: Yes, you can simplify an expression with multiple exponents by applying the rules of exponents. For example, $\frac{x^2}{x^3} \times x^4 = x^{2-3+4} = x^3$ .

In conclusion, simplifying exponents is an essential skill in mathematics that can be applied to a wide range of problems. By understanding the rules of exponents and practicing simplifying expressions, you can become more confident and proficient in solving mathematical problems. We hope this Q&A article has provided you with a better understanding of simplifying exponents and has addressed any questions or concerns you may have had.

Always apply the rules of exponents in the correct order.
Be careful when simplifying expressions with negative exponents.
Practice simplifying expressions with multiple exponents.
Use the order of operations to simplify complex expressions.

The final answer is $\boxed{\frac{13}{12}}$ .