Problem 103
Question
Simplify the expression and write it with rational exponents. Assume that all variables are positive. $$ \mathbf{p}^{1 / 2}\left(\mathbf{p}^{3 / 2}+\mathbf{p}^{\mathbf{1} / 2}\right) $$
Step-by-Step Solution
Verified Answer
\( \mathbf{p}^2 + \mathbf{p} \)
1Step 1: Distribute the Expression
Expand the expression by distributing \( \mathbf{p}^{1 / 2} \) to every term inside the parentheses: \( \mathbf{p}^{1 / 2} \times \mathbf{p}^{3 / 2} + \mathbf{p}^{1 / 2} \times \mathbf{p}^{1 / 2} \).
2Step 2: Apply the Product of Powers Rule
Apply the product of powers rule, which states \( a^m \times a^n = a^{m+n} \), to each term in the expression:- For \( \mathbf{p}^{1 / 2} \times \mathbf{p}^{3 / 2} \), we get \( \mathbf{p}^{(1/2 + 3/2)} = \mathbf{p}^2 \).- For \( \mathbf{p}^{1 / 2} \times \mathbf{p}^{1 / 2} \), we get \( \mathbf{p}^{(1/2 + 1/2)} = \mathbf{p}^1 \).
3Step 3: Simplify the Expression
Combine the results from Step 2. The simplified expression is \( \mathbf{p}^2 + \mathbf{p}^1 \). Write the final answer as \( \mathbf{p}^2 + \mathbf{p} \).
Key Concepts
Product of Powers RuleSimplifying ExpressionsAlgebraic Expressions
Product of Powers Rule
When working with exponents, the Product of Powers Rule is a helpful tool. This rule simplifies expressions where the same base number is multiplied by itself with different exponents. If you have an expression like \( a^m \times a^n \), the rule states that you can simply add the exponents together. So, it becomes \( a^{m+n} \).
This rule is particularly useful when dealing with algebraic expressions involving exponents, allowing you to combine terms neatly. In our example, \( \mathbf{p}^{1/2} \times \mathbf{p}^{3/2} \) was simplified using this rule to become \( \mathbf{p}^{2} \).
Remember:
This rule is particularly useful when dealing with algebraic expressions involving exponents, allowing you to combine terms neatly. In our example, \( \mathbf{p}^{1/2} \times \mathbf{p}^{3/2} \) was simplified using this rule to become \( \mathbf{p}^{2} \).
Remember:
- The bases must be the same for this rule to apply.
- Only add the exponents; do not change the base.
Simplifying Expressions
Simplifying an expression means to rewrite it in a way that is easier to work with, without changing the expression's value. The goal is to make the expression as concise as possible while preserving its meaning.
For instance, if you have a complex expression inside parentheses, you might start simplifying by expanding or distributing any terms outside the parentheses across those inside. In our exercise, \( \mathbf{p}^{1/2} \) was distributed across each term inside the parentheses, resulting in two separate terms: \( \mathbf{p}^{1/2} \times \mathbf{p}^{3/2} \) and \( \mathbf{p}^{1/2} \times \mathbf{p}^{1/2} \).
Once distributed, each product of powers was further simplified using the Product of Powers Rule.
Key steps in simplification:
For instance, if you have a complex expression inside parentheses, you might start simplifying by expanding or distributing any terms outside the parentheses across those inside. In our exercise, \( \mathbf{p}^{1/2} \) was distributed across each term inside the parentheses, resulting in two separate terms: \( \mathbf{p}^{1/2} \times \mathbf{p}^{3/2} \) and \( \mathbf{p}^{1/2} \times \mathbf{p}^{1/2} \).
Once distributed, each product of powers was further simplified using the Product of Powers Rule.
Key steps in simplification:
- Distribute and expand terms as needed.
- Apply the appropriate mathematical rules, like the Product of Powers Rule.
- Combine like terms for a cleaner expression.
Algebraic Expressions
Algebraic expressions are combinations of letters and numbers that stand for specific values or operations. They often include numbers (constants), variables (symbols that stand for numbers), and mathematical operations like addition, subtraction, multiplication, and division.
Variables, such as \( \mathbf{p} \) in our example, are used to represent numbers that might change or aren't clearly defined at the outset. Algebra enables us to solve real-world problems by setting up equations that express relationships between different quantities.
In the given exercise, the expression \( \mathbf{p}^{1 / 2}\left(\mathbf{p}^{3 / 2}+\mathbf{p}^{\mathbf{1} / 2}\right) \) is an example of an algebraic expression involving rational exponents. Here, rational exponents show fractional powers, implying roots—the reciprocal of these fractions acts as roots (\( \frac{1}{2} \) implies square root).
Variables, such as \( \mathbf{p} \) in our example, are used to represent numbers that might change or aren't clearly defined at the outset. Algebra enables us to solve real-world problems by setting up equations that express relationships between different quantities.
In the given exercise, the expression \( \mathbf{p}^{1 / 2}\left(\mathbf{p}^{3 / 2}+\mathbf{p}^{\mathbf{1} / 2}\right) \) is an example of an algebraic expression involving rational exponents. Here, rational exponents show fractional powers, implying roots—the reciprocal of these fractions acts as roots (\( \frac{1}{2} \) implies square root).
- Simplifying algebraic expressions helps in evaluating them effectively.
- Understanding terms and their interactions are crucial for working with larger algebraic equations.
Other exercises in this chapter
Problem 103
Rationalize the denominator. $$ \frac{1}{3-\sqrt{2}} $$
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Factor the expression completely. \(16 x^{2}-25\)
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Clear fractions and solve. $$ \frac{1}{x}+\frac{3}{x^{2}}=0 $$
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Rationalize the denominator. $$ \frac{1}{\sqrt{3}-2} $$
View solution