Problem 43 Use the rules of exponents to si... [FREE SOLUTION]

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Chapter 7: Problem 43

Use the rules of exponents to simplify expression. \(3^{1 / 3} 3^{1 / 4}\)

Short Answer

Expert verified

The simplified expression is \(3^{7/12}\).

Step by step solution

- Identify the base

Notice that both terms in the expression, \(3^{1 / 3}\) and \(3^{1 / 4}\), have the same base, which is 3.

- Apply the exponent addition rule

When multiplying like bases, add the exponents together. The rule states \(a^m \times a^n = a^{m+n}\). Thus, \(3^{1/3} \times 3^{1/4} = 3^{1/3 + 1/4}\).

- Find a common denominator

To add the fractions \(1/3\) and \(1/4\), find a common denominator. The least common denominator for 3 and 4 is 12. Therefore, rewrite the fractions: \(1/3 = 4/12\) and \(1/4 = 3/12\).

- Add the fractions

Now add the fractions with the common denominator: \(4/12 + 3/12 = 7/12\).

- Simplify the exponent

Combine the exponents: \(3^{1/3 + 1/4} = 3^{7/12}\). This is the simplified form of the expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponent Addition

When dealing with exponents, learning how to add them is essential. The rule to remember is when you multiply two expressions with the same base, you add their exponents.
This is known as the exponent addition rule, written as \(a^m \times a^n = a^{m+n}\).
For example, if you have \(3^{1/3} \times 3^{1/4}\), you add the exponents \(\frac{1}{3} + \frac{1}{4}\).
Adding fractions requires a common denominator, which leads us to our next concept.

Common Denominators

Adding fractions involves finding a common denominator. This is a shared number that both denominators can divide into evenly.
For instance, with \(\frac{1}{3} + \frac{1}{4}\), the least common denominator is 12.
Then, rewrite \(\frac{1}{3}\) as \(\frac{4}{12}\) and \(\frac{1}{4}\) as \(\frac{3}{12}\).
Now, it's easy to add them: \(\frac{4}{12} + \frac{3}{12} = \frac{7}{12}\).
This new fraction can be added, and we can use it in our next step in simplifying the expression.

Simplifying Expressions

After adding the fractions, simplify the expression. In our example, multiplying the bases \(3^{1/3} \times 3^{1/4}\) results in adding exponents: \(3^{1/3 + 1/4} = 3^{7/12}\).
This is the simplified form.
Remember to always reduce your expressions to their simplest form for clarity.

Identify the base.
Apply exponent addition.
Find common denominators for any fractional exponents.
Add fractions and simplify.

Following these steps ensures clarity and accuracy in your solutions.

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91影视

Use the rules of exponents to simplify expression. \(3^{1 / 3} 3^{1 / 4}\)

Short Answer

Step by step solution

- Identify the base

- Apply the exponent addition rule

- Find a common denominator

- Add the fractions

- Simplify the exponent

Key Concepts

Exponent Addition

Common Denominators

Simplifying Expressions

One App. One Place for Learning.

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