Simplify. Assume All Variables Are Positive. X 7 6 ⋅ X 5 4 X^{\frac{7}{6}} \cdot X^{\frac{5}{4}} X 6 7 ​ ⋅ X 4 5 ​ Write Your Answer In The Form A A A Or A B \frac{A}{B} B A ​ Where A A A And B B B Are Constants Or Variable Expressions That Have No

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Understanding the Problem

When dealing with exponents, it's essential to remember the properties of exponentiation. In this problem, we are given the expression $x^{\frac{7}{6}} \cdot x^{\frac{5}{4}}$, and we need to simplify it. The expression involves two variables with fractional exponents, and our goal is to combine them into a single expression.

Properties of Exponentiation

Before we dive into simplifying the expression, let's recall some essential properties of exponentiation:

• Product of Powers Property: When multiplying two powers with the same base, we add the exponents. Mathematically, this can be represented as $a^m \cdot a^n = a^{m+n}$.
• Power of a Power Property: When raising a power to another power, we multiply the exponents. Mathematically, this can be represented as $(a^m)^n = a^{m \cdot n}$.

Simplifying the Expression

Now that we have a solid understanding of the properties of exponentiation, let's apply them to simplify the given expression.

We can rewrite the expression as $x^{\frac{7}{6}} \cdot x^{\frac{5}{4}} = x^{\frac{7}{6} + \frac{5}{4}}$. This is based on the product of powers property, where we add the exponents.

Finding a Common Denominator

To add the exponents, we need to find a common denominator. The least common multiple (LCM) of 6 and 4 is 12. We can rewrite the fractions with a common denominator as follows:

$\frac{7}{6} = \frac{7 \cdot 2}{6 \cdot 2} = \frac{14}{12}$

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

Adding the Exponents

Now that we have a common denominator, we can add the exponents:

$\frac{14}{12} + \frac{15}{12} = \frac{29}{12}$

Simplifying the Expression

We can now rewrite the expression as $x^{\frac{29}{12}}$. This is the simplified form of the given expression.

Conclusion

In this article, we simplified the expression $x^{\frac{7}{6}} \cdot x^{\frac{5}{4}}$ using the properties of exponentiation. We applied the product of powers property to add the exponents and found a common denominator to simplify the expression. The final answer is $x^{\frac{29}{12}}$.

Frequently Asked Questions

• What is the product of powers property? The product of powers property states that when multiplying two powers with the same base, we add the exponents. Mathematically, this can be represented as $a^m \cdot a^n = a^{m+n}$.
• What is the power of a power property? The power of a power property states that when raising a power to another power, we multiply the exponents. Mathematically, this can be represented as $(a^m)^n = a^{m \cdot n}$.
• How do I simplify an expression with fractional exponents? To simplify an expression with fractional exponents, you need to find a common denominator and add the exponents. You can use the product of powers property to add the exponents and the power of a power property to simplify the expression.

Step-by-Step Solution

Here's a step-by-step solution to simplify the expression $x^{\frac{7}{6}} \cdot x^{\frac{5}{4}}$:

1. Apply the product of powers property: Rewrite the expression as $x^{\frac{7}{6} + \frac{5}{4}}$.
2. Find a common denominator: Find the least common multiple (LCM) of 6 and 4, which is 12.
3. Rewrite the fractions with a common denominator: Rewrite the fractions as $\frac{14}{12}$ and $\frac{15}{12}$.
4. Add the exponents: Add the exponents to get $\frac{29}{12}$.
5. Simplify the expression: Rewrite the expression as $x^{\frac{29}{12}}$.

Final Answer

The final answer is $x^{\frac{29}{12}}$.

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Understanding the Problem

When dealing with exponents, it's essential to remember the properties of exponentiation. In this problem, we are given the expression $x^{\frac{7}{6}} \cdot x^{\frac{5}{4}}$, and we need to simplify it. The expression involves two variables with fractional exponents, and our goal is to combine them into a single expression.

Properties of Exponentiation

Before we dive into simplifying the expression, let's recall some essential properties of exponentiation:

• Product of Powers Property: When multiplying two powers with the same base, we add the exponents. Mathematically, this can be represented as $a^m \cdot a^n = a^{m+n}$.
• Power of a Power Property: When raising a power to another power, we multiply the exponents. Mathematically, this can be represented as $(a^m)^n = a^{m \cdot n}$.

Simplifying the Expression

Now that we have a solid understanding of the properties of exponentiation, let's apply them to simplify the given expression.

We can rewrite the expression as $x^{\frac{7}{6}} \cdot x^{\frac{5}{4}} = x^{\frac{7}{6} + \frac{5}{4}}$. This is based on the product of powers property, where we add the exponents.

Finding a Common Denominator

To add the exponents, we need to find a common denominator. The least common multiple (LCM) of 6 and 4 is 12. We can rewrite the fractions with a common denominator as follows:

$\frac{7}{6} = \frac{7 \cdot 2}{6 \cdot 2} = \frac{14}{12}$

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

Adding the Exponents

Now that we have a common denominator, we can add the exponents:

$\frac{14}{12} + \frac{15}{12} = \frac{29}{12}$

Simplifying the Expression

We can now rewrite the expression as $x^{\frac{29}{12}}$. This is the simplified form of the given expression.

Conclusion

In this article, we simplified the expression $x^{\frac{7}{6}} \cdot x^{\frac{5}{4}}$ using the properties of exponentiation. We applied the product of powers property to add the exponents and found a common denominator to simplify the expression. The final answer is $x^{\frac{29}{12}}$.

Frequently Asked Questions

Q: What is the product of powers property?

A: The product of powers property states that when multiplying two powers with the same base, we add the exponents. Mathematically, this can be represented as $a^m \cdot a^n = a^{m+n}$.

Q: What is the power of a power property?

A: The power of a power property states that when raising a power to another power, we multiply the exponents. Mathematically, this can be represented as $(a^m)^n = a^{m \cdot n}$.

Q: How do I simplify an expression with fractional exponents?

A: To simplify an expression with fractional exponents, you need to find a common denominator and add the exponents. You can use the product of powers property to add the exponents and the power of a power property to simplify the expression.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. For example, the LCM of 6 and 4 is 12.

Q: How do I find a common denominator?

A: To find a common denominator, you need to find the least common multiple (LCM) of the denominators. For example, to find a common denominator for $\frac{7}{6}$ and $\frac{5}{4}$, you need to find the LCM of 6 and 4, which is 12.

Q: How do I add fractions with different denominators?

A: To add fractions with different denominators, you need to find a common denominator and add the numerators. For example, to add $\frac{7}{6}$ and $\frac{5}{4}$, you need to find a common denominator, which is 12, and add the numerators: $\frac{14}{12} + \frac{15}{12} = \frac{29}{12}$.

Step-by-Step Solution

Here's a step-by-step solution to simplify the expression $x^{\frac{7}{6}} \cdot x^{\frac{5}{4}}$:

1. Apply the product of powers property: Rewrite the expression as $x^{\frac{7}{6} + \frac{5}{4}}$.
2. Find a common denominator: Find the least common multiple (LCM) of 6 and 4, which is 12.
3. Rewrite the fractions with a common denominator: Rewrite the fractions as $\frac{14}{12}$ and $\frac{15}{12}$.
4. Add the exponents: Add the exponents to get $\frac{29}{12}$.
5. Simplify the expression: Rewrite the expression as $x^{\frac{29}{12}}$.

Final Answer

The final answer is $x^{\frac{29}{12}}$.