Problem 12 - Contemporary Precalculus

Question

Compute and simplify. $$x^{1 / 2}\left(3 x^{3 / 2}+2 x^{-1 / 2}\right)$$

Step-by-Step Solution

Answer

Question: Simplify the following expression: \(x^{1/2}(3x^{3/2} + 2x^{-1/2})\) Answer: The simplified expression is \(3x^2 + 2\).

Step 1: Distribute the term outside the parentheses to the terms within the parentheses.

To do this, we will need to multiply \(x^{1/2}\) by each term inside the parentheses: \(3x^{3/2}\) and \(2x^{-1/2}\). We will apply the exponent rules when multiplying these expressions.

Step 2: Apply exponent rules

When multiplying expressions with the same base, we add the exponents: \(a^m \times a^n = a^{m+n}\). So, for both terms: 1. \(x^{1/2} \times 3x^{3/2} = 3x^{(1/2)+(3/2)}\) 2. \(x^{1/2} \times 2x^{-1/2} = 2x^{(1/2)+(-1/2)}\)

Step 3: Simplify the expressions

Now, let's simplify the expressions further by calculating the new exponents: 1. \(3x^{(1/2)+(3/2)} = 3x^{(4/2)}\) 2. \(2x^{(1/2)+(-1/2)} = 2x^{0}\) Recall that any non-zero number raised to the power of 0 is 1. So, \(2x^0 = 2(1) = 2\). Therefore, the simplified expression is: $$3x^{(4/2)} + 2$$

Step 4: Present the final simplified expression

Putting everything together, the simplified expression is: $$3x^2 + 2$$

Key Concepts

Exponent Rules

When working with exponents, it's crucial to understand how they behave during multiplication. One of the key rules for exponents is that when you multiply two powers with the same base, you simply add their exponents together. This is represented by the formula: \(a^m imes a^n = a^{m+n}\).
For example, in the exercise provided, we have powers of \(x\): \(x^{1/2}\) and \(x^{3/2}\). When multiplying them, add the exponents: \((1/2) + (3/2) = 4/2\), which simplifies to \(2\).
This rule simplifies complicated expressions by reducing the number of terms with exponents. It's important to ensure the base of the powers are identical to apply this rule.

• The expression \(x^{1/2} \times 3x^{3/2} = 3x^2\) demonstrates this rule.
• Similarly, \(x^{1/2} \times 2x^{-1/2} = 2x^0\) shows how exponents cancel each other out.

Once the exponents are added and simplified, the expressions themselves become much easier to handle.

Distributive Property

The Distributive Property is a fundamental principle used in various types of algebraic operations, including polynomial simplification. This property tells us how to multiply a single term by each term inside a parenthesis.
For example, in the given expression \(x^{1/2}(3x^{3/2} + 2x^{-1/2})\), the Distributive Property requires you to distribute \(x^{1/2}\) to every term inside the parentheses. This means multiplying \(x^{1/2}\) with both \(3x^{3/2}\) and \(2x^{-1/2}\):

• Multiply \(x^{1/2} \times 3x^{3/2}\).
• Multiply \(x^{1/2} \times 2x^{-1/2}\).

By doing this distribution, you maintain equality throughout the expression and prepare the terms for simplification using other algebraic rules, such as the Exponent Rules.
This property not only works with algebraic terms but can also be applied in real numbers arithmetic to simplify addition and subtraction after multiplication.

Simplifying Expressions

Simplifying expressions is all about reducing them to their simplest form, making them easier to understand and work with. In our example, after applying the Distributive Property and Exponent Rules, the expression becomes easier to handle, transitioning from a complex series of multiplications to a straightforward sum.
After applying the exponent operations, the expression \(3x^{4/2} + 2x^0\) reduces to simpler forms because:

• \(3x^{4/2} = 3x^2\).
• \(2x^0\) simplifies to \(2\) because any non-zero number raised to the power of zero is \(1\). Thus, \(2x^0\) becomes \(2\).

The final step is the combination of these terms into \(3x^2 + 2\).
This is the effortlessly understood and used form of the expression, ready for any further calculations or evaluations that might be necessary. Understanding and applying these simplification steps is key to mastering more complex algebra problems.