Understanding Exponent Rules

Oct 13, 2025 by recoveryshake.com 138 views

**Simplifying Exponents: A Guide to Finding the Product with the Exponent in Simplest Form**

What is an Exponent?

An exponent is a small number that is written above and to the right of a number or a variable. It represents the power to which the base number or variable is raised. For example, in the expression $6^{\frac{1}{3}}$ , the exponent $\frac{1}{3}$ indicates that the base number 6 is raised to the power of $\frac{1}{3}$ .

Understanding Exponent Rules

To simplify exponents, we need to understand the rules of exponents. The two main rules are:

Product of Powers Rule: When multiplying two numbers with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m+n}$ .
Power of a Power Rule: When raising a number with an exponent to another power, we multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$ .

Simplifying Exponents with Fractions

Now, let's simplify the expression $6^{\frac{1}{3}} \cdot 6^{\frac{1}{4}}$ . Using the product of powers rule, we add the exponents:

6^{\frac{1}{3}} \cdot 6^{\frac{1}{4}} = 6^{\frac{1}{3} + \frac{1}{4}}

To add the fractions, we need a common denominator, which is 12. So, we rewrite the fractions with a common denominator:

6^{\frac{4}{12} + \frac{3}{12}} = 6^{\frac{7}{12}}

Therefore, the product of $6^{\frac{1}{3}}$ and $6^{\frac{1}{4}}$ is $6^{\frac{7}{12}}$ .

Finding the Values of x and y

Now, let's identify the values of $x$ and $y$ in the expression $6^{\frac{x}{y}}$ . We know that the product of $6^{\frac{1}{3}}$ and $6^{\frac{1}{4}}$ is $6^{\frac{7}{12}}$ . Therefore, we can set up the equation:

6^{\frac{x}{y}} = 6^{\frac{7}{12}}

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

\frac{x}{y} = \frac{7}{12}

To find the values of $x$ and $y$ , we can cross-multiply:

12x = 7y

Now, we need to find two numbers whose product is 84 (12x) and whose ratio is 7:12. The numbers are 84 and 12. Therefore, we can set up the equations:

x = 84

y = 12

Conclusion

In this article, we simplified the expression $6^{\frac{1}{3}} \cdot 6^{\frac{1}{4}}$ using the product of powers rule and found the values of $x$ and $y$ in the expression $6^{\frac{x}{y}}$ . We learned that the product of $6^{\frac{1}{3}}$ and $6^{\frac{1}{4}}$ is $6^{\frac{7}{12}}$ and that the values of $x$ and $y$ are 84 and 12, respectively.

Frequently Asked Questions

Q: What is an exponent?

A: An exponent is a small number that is written above and to the right of a number or a variable. It represents the power to which the base number or variable is raised.

Q: What are the rules of exponents?

A: The two main rules of exponents are the product of powers rule and the power of a power rule. The product of powers rule states that when multiplying two numbers with the same base, we add the exponents. The power of a power rule states that when raising a number with an exponent to another power, we multiply the exponents.

Q: How do I simplify exponents with fractions?

A: To simplify exponents with fractions, we need to add the fractions by finding a common denominator. We can then rewrite the fractions with a common denominator and add the numerators.

Q: How do I find the values of x and y in the expression $6^{\frac{x}{y}}$ ?

A: To find the values of x and y, we need to equate the exponents of the two expressions. We can then cross-multiply and solve for x and y.

Q: What are the values of x and y in the expression $6^{\frac{x}{y}}$ ?

A: The values of x and y are 84 and 12, respectively.

Q: How do I apply the product of powers rule?

A: To apply the product of powers rule, we need to multiply two numbers with the same base. We then add the exponents.

Q: How do I apply the power of a power rule?

A: To apply the power of a power rule, we need to raise a number with an exponent to another power. We then multiply the exponents.

Q: What is the product of $6^{\frac{1}{3}}$ and $6^{\frac{1}{4}}$ ?

A: The product of $6^{\frac{1}{3}}$ and $6^{\frac{1}{4}}$ is $6^{\frac{7}{12}}$ .