Understanding Exponents and Their Properties

Oct 13, 2025 by magicalceylontours.com 74 views

Understanding Exponents and Their Properties

In mathematics, exponents are a fundamental concept used to represent repeated multiplication of a number. For instance, $6^3$ means $6$ multiplied by itself $3$ times, which equals $216$ . Exponents can also be expressed as fractional numbers, such as $6^{\frac{1}{3}}$ , which represents the cube root of $6$ . In this article, we will explore the concept of simplifying exponents and apply it to a specific problem.

The Product of Powers Property

One of the essential properties of exponents is the product of powers property, which states that when multiplying two numbers with the same base, we can add their exponents. Mathematically, this can be represented as:

$a^m \cdot a^n = a^{m+n}$

where $a$ is the base and $m$ and $n$ are the exponents.

Applying the Product of Powers Property

Now, let's apply this property to the given problem:

$6^{\frac{1}{3}} \cdot 6^{\frac{1}{4}} = 6^{\frac{x}{4}}$

Using the product of powers property, we can simplify the left-hand side of the equation by adding the exponents:

$6^{\frac{1}{3} + \frac{1}{4}} = 6^{\frac{x}{4}}$

To add the fractions, we need to find a common denominator, which is $12$ in this case:

$6^{\frac{4}{12} + \frac{3}{12}} = 6^{\frac{x}{4}}$

Simplifying the exponents, we get:

$6^{\frac{7}{12}} = 6^{\frac{x}{4}}$

Equating the Exponents

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

$\frac{7}{12} = \frac{x}{4}$

To solve for $x$ , we can cross-multiply:

$7 \cdot 4 = 12 \cdot x$

Simplifying the equation, we get:

$28 = 12x$

Dividing both sides by $12$ , we get:

$x = \frac{28}{12}$

Simplifying the fraction, we get:

$x = \frac{7}{3}$

Finding the Value of $y$

Now that we have found the value of $x$ , we can find the value of $y$ by substituting $x$ into the original equation:

$6^{\frac{1}{3}} \cdot 6^{\frac{1}{4}} = 6^{\frac{x}{4}}$

Substituting $x = \frac{7}{3}$ , we get:

$6^{\frac{1}{3}} \cdot 6^{\frac{1}{4}} = 6^{\frac{7}{12}}$

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

$\frac{1}{3} + \frac{1}{4} = \frac{7}{12}$

Simplifying the equation, we get:

$\frac{4}{12} + \frac{3}{12} = \frac{7}{12}$

Simplifying the fractions, we get:

$\frac{7}{12} = \frac{7}{12}$

This confirms that the value of $y$ is indeed $\frac{7}{12}$ .

Conclusion

In this article, we have explored the concept of simplifying exponents and applied it to a specific problem. We have used the product of powers property to simplify the left-hand side of the equation and equated the exponents to find the values of $x$ and $y$ . The final answer is:

$x = \frac{7}{3}$ , $y = \frac{7}{12}$

Additional Tips and Tricks

When simplifying exponents, make sure to use the product of powers property to combine the exponents.
When equating the exponents, make sure to use the same base on both sides of the equation.
When solving for $x$ or $y$ , make sure to simplify the fractions and decimals to their simplest form.

By following these tips and tricks, you can become more confident in simplifying exponents and solving equations involving exponents.