Problem 51 - Precalculus Mathematics for Calculus

Question

Perform the indicated operations and simplify. $$y^{1 / 3}\left(y^{2 / 3}+y^{5 / 3}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \( y + y^2 \).

Step 1: Distribute the Term

Distribute the term \( y^{1/3} \) to each term within the parentheses. This involves multiplying \( y^{1/3} \) with \( y^{2/3} \) and \( y^{1/3} \) with \( y^{5/3} \).

Step 2: Apply the Product of Powers Rule

Use the property that when multiplying like bases you add the exponents, i.e., \( a^m \times a^n = a^{m+n} \). First, compute \( y^{1/3} \times y^{2/3} = y^{(1/3 + 2/3)} = y^1 = y \).

Step 3: Continue with the Second Term

Similarly, compute \( y^{1/3} \times y^{5/3} = y^{(1/3 + 5/3)} = y^2 \).

Step 4: Write the Simplified Expression

After performing the operations, the expression simplifies to \( y + y^2 \). This is the final simplified form.

Key Concepts

Product of Powers

The **product of powers** rule is a fundamental concept in algebra that allows us to simplify expressions with exponents. It states that when you multiply two exponential terms with the same base, you can add their exponents. This rule can be expressed mathematically as \[a^m \times a^n = a^{m+n}.\]This is a handy rule because it simplifies the process of working with exponents. In our exercise, we used this rule to simplify the expression\[y^{1/3} \times y^{2/3} = y^{(1/3 + 2/3)} = y^1 = y.\]Another example given is \[y^{1/3} \times y^{5/3} = y^{(1/3 + 5/3)} = y^2.\]When simplifying, make sure the bases (in this case, 'y') are the same. This rule greatly simplifies expressions into more manageable forms. The product of powers rule helps streamline complex multiplication, making calculations much easier and preventing errors.

Simplifying Expressions

**Simplifying expressions** involves reducing complex algebraic expressions to their simplest form. It often uses rules like the product of powers, distributive property, and combining like terms. By simplifying expressions, we aim to make them easier to understand and work with.In our exercise, the simplification starts by distributing the term \( y^{1/3} \) inside the parentheses and applying the product of powers rule:

• \( y^{1/3} \times y^{2/3} = y \)
• \( y^{1/3} \times y^{5/3} = y^2 \)

The expression reduces from a fractional exponent format to\[y + y^2.\]Simplifying is crucial in algebra to help tidy up calculations and provide a clear solution. It ensures you are always working with the most accessible version of mathematical expressions. This process often reveals elegant and simple forms of more complex beginnings.

Algebraic Operations

Understanding **algebraic operations** is key to dealing with algebraic expressions. These operations include addition, subtraction, multiplication, division, and exponentiation. Each has specific rules and properties that make working with expressions efficient and systematic.In the exercise, we used two primary operations—multiplication and addition:

• **Multiplication**: Distribute \( y^{1/3} \) over each term inside the parentheses, which is a basic operation when simplifying expressions.
• **Addition**: After simplifying terms, add them via simple addition: \[y + y^2. \]

It demonstrates how multiplication can transform the initial expression, while addition compiles the results into a neat and understandable format.In algebra, operations often work together in a sequence, applying one operation after another to systematically simplify or solve an equation or expression. Knowing how to execute each operation effectively and in the right order is essential to mastering algebra.