Problem 63 Simplify each expression. Assume... [FREE SOLUTION]

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Chapter 10: Problem 63

Simplify each expression. Assume that all variables represent positive real numbers. $$ 3^{1 / 2} \cdot 3^{3 / 2} $$

Short Answer

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Step by step solution

Identify the Problem

Simplify the expression oinvolving exponents: oinvolving exponents: $$ 3^{1 / 2} \times 3^{3 / 2} $$.

Apply the Product of Powers Rule

The product of powers rule states that when multiplying two exponents with the same base, you add the exponents: oformula: $$ a^m \times a^n = a^{m+n} $$. Apply this rule to the given expression.

Add the Exponents

Add the exponents oformula: $$ 3^{1/2} \times 3^{3/2} = 3^{(1/2 + 3/2)} $$. The sum of the exponents is: oformula: $$ 1/2 + 3/2 = 4/2 = 2 $$.

Simplify the Final Expression

Simplify the exponent: oformula: $$ 3^2 $$, which equals 9.

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Key Concepts

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

Product of Powers Rule

The Product of Powers Rule is a fundamental concept when working with exponents. This rule states that when you multiply two expressions with the same base, you should add their exponents. For example, with a base of 3, the expressions can be written as $3^m$ and $3^n$. According to the rule, this multiplication simplifies to $3^{m+n}$. In our original exercise, we have $3^{1 / 2} \times 3^{3 / 2}$. The base is the same, which is 3. Using the Product of Powers Rule, we add the exponents: $\frac{1}{2} + \frac{3}{2}$. This helps simplify the expression easily.

Adding Exponents

Adding exponents is a critical step in simplifying expressions with the same base. When you have an expression like $3^{1/2} \times 3^{3/2}$, you follow the Product of Powers Rule by adding the exponents. Here's a detailed breakdown:

First, identify the exponents to add: $\frac{1}{2}$ and $\frac{3}{2}$.
Next, perform the addition: $\frac{1}{2} + \frac{3}{2} = \frac{4}{2}$.
Simplify the resulting fraction: $\frac{4}{2} = 2$.

By adding the exponents, we convert the expression to a simpler form: $3^2$.

Simplified Exponents

Once you have added the exponents and simplified the fraction, the last step is to simplify the final expression. From our example, $3^{1 / 2} \times 3^{3 / 2}$ turns into $3^2$ after adding the exponents. Simplifying the exponent means calculating the value of $3^2$.

$3^2$ means multiplying 3 by itself: $3 \times 3$.
This results in 9.

So, the fully simplified expression of the original problem $3^{1/2} \times 3^{3/2}$ is 9. By following these steps鈥攗sing the Product of Powers Rule, adding exponents, and simplifying them鈥攜ou can handle most expressions with exponents confidently.

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91影视

Simplify each expression. Assume that all variables represent positive real numbers. $$ 3^{1 / 2} \cdot 3^{3 / 2} $$

Short Answer

Step by step solution

Identify the Problem

Apply the Product of Powers Rule

Add the Exponents

Simplify the Final Expression

Key Concepts

Product of Powers Rule

Adding Exponents

Simplified Exponents

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