Problem 61
Question
Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers. $$ \frac{y^{3} w^{10}}{y w^{5}} $$
Step-by-Step Solution
Verified Answer
Question: Simplify the expression using the product rule and quotient rule of exponents: $\frac{y^{3} w^{10}}{y w^{5}}$
Answer: The simplified expression is $y^2w^5$.
1Step 1: Identify the given expression
The given expression is:
$$
\frac{y^{3} w^{10}}{y w^{5}}
$$
2Step 2: Apply the quotient rule of exponents
The quotient rule of exponents states that:
$$
x^{a} \div x^{b} = x^{a-b}
$$
Using the quotient rule, we can simplify the given expression as follows:
$$
\frac{y^{3} w^{10}}{y w^{5}} = \frac{y^{3}}{y} \times \frac{w^{10}}{w^5}
$$
3Step 3: Simplify y terms using the quotient rule
Applying the quotient rule of exponents to the y terms:
$$
\frac{y^{3}}{y} = y^{3-1} = y^2
$$
4Step 4: Simplify w terms using the quotient rule
Applying the quotient rule of exponents to the w terms:
$$
\frac{w^{10}}{w^5} = w^{10-5} = w^5
$$
5Step 5: Write the simplified expression
Combine the simplified y and w terms to obtain the final simplified expression:
$$
y^2 \times w^5 = y^2w^5
$$
The simplified expression is:
$$
y^2w^5
$$
Key Concepts
Product RuleQuotient RuleSimplificationAlgebraic Expressions
Product Rule
The product rule of exponents is a helpful tool when multiplying expressions that have the same base. When you're dealing with exponents, remember this useful rule: when multiplying two exponents with the same base, you simply add the exponents together. For instance, for a base \(x\), the rule states that \(x^a \times x^b = x^{a+b}\).
This makes it easy to combine terms and simplify expressions. By using the product rule, multiple expressions with the same base can be simplified into a single term, making calculations more manageable.
This rule is fundamental in simplifying expressions and helps in keeping equations neat and concise.
This makes it easy to combine terms and simplify expressions. By using the product rule, multiple expressions with the same base can be simplified into a single term, making calculations more manageable.
This rule is fundamental in simplifying expressions and helps in keeping equations neat and concise.
Quotient Rule
The quotient rule of exponents is crucial when simplifying expressions where one exponent is divided by another. This rule states that when you divide two exponents with the same base, you subtract the exponent of the denominator from the exponent of the numerator.
Here's how it works: for the base \(x\), \( \frac{x^a}{x^b} = x^{a-b}\). This property is particularly useful for reducing the complexity of algebraic expressions.
Here's how it works: for the base \(x\), \( \frac{x^a}{x^b} = x^{a-b}\). This property is particularly useful for reducing the complexity of algebraic expressions.
- Apply this rule when you have terms with the same base in both the numerator and the denominator.
- Subtracting exponents allows you to simplify expressions step-by-step until no further simplification is possible.
Simplification
Simplification is the process of reducing expressions to their simplest form, making them easier to work with. In algebra, simplification involves using rules like the product and quotient rules to combine and reduce terms.
During simplification, you aim to make the expression as clear and straightforward as possible, often by reducing the number of terms or layers of complexity. This process is essential because simplified expressions are easier to manipulate, interpret, and use in further calculations.
During simplification, you aim to make the expression as clear and straightforward as possible, often by reducing the number of terms or layers of complexity. This process is essential because simplified expressions are easier to manipulate, interpret, and use in further calculations.
- Using simplification, unnecessary steps or components can be removed.
- The goal is to achieve the simplest equivalent expression.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation symbols. Understanding algebraic expressions involves learning how to manipulate them using various algebraic rules to simplify or solve them.
An algebraic expression like \( \frac{y^{3} w^{10}}{y w^{5}} \) involves both variables and exponents, and simplifying these requires careful application of the rules of exponents. These expressions can represent real-world quantities and allow us to generalize arithmetic operations.
An algebraic expression like \( \frac{y^{3} w^{10}}{y w^{5}} \) involves both variables and exponents, and simplifying these requires careful application of the rules of exponents. These expressions can represent real-world quantities and allow us to generalize arithmetic operations.
- Algebraic expressions are essential for solving equations and inequalities.
- They are the building blocks of algebra, helping to model and solve problems.
Other exercises in this chapter
Problem 60
The number of different ways 5 people can be arranged in a row is \(5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 .\) How many ways is this?
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Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$
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Use the order of operations to simplify the quantities for the following problems. $$ (7)(16)-9^{2}+4\left(1^{1}+3^{2}\right) $$
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