Problem 77 - Algebra and Trigonometry

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 Expression Inside the Parentheses

Start by distributing the term outside of the parentheses, \( y^{1/3} \), to each term within the parentheses, which are \( y^{2/3} \) and \( y^{5/3} \).

Step 2: Apply the Power of a Power Property

Use the property \( a^m \times a^n = a^{m+n} \) for the distribution. First, apply it to \( y^{1/3} \cdot y^{2/3} \): \[ y^{1/3} \cdot y^{2/3} = y^{(1/3) + (2/3)} = y^{3/3} = y^1 = y \]

Step 3: Continue Applying the Power Property

Next, apply the property to the second term inside the parentheses: \( y^{1/3} \cdot y^{5/3} \). This gives:\[ y^{1/3} \cdot y^{5/3} = y^{(1/3) + (5/3)} = y^{6/3} = y^2 \]

Step 4: Write the Resulting Expression

Combine the results from Step 2 and Step 3. The expression simplifies down to:\[ y + y^2 \]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by each term inside a set of parentheses. By doing so, it simplifies expressions and makes it easier to find the final result. In this exercise, we have an expression \( y^{1/3}(y^{2/3} + y^{5/3}) \), where the term \( y^{1/3} \) outside the parentheses needs to be distributed to each term inside the parentheses—\( y^{2/3} \) and \( y^{5/3} \).

• Distribute: Multiply \( y^{1/3} \) by \( y^{2/3} \)


• Distribute: Multiply \( y^{1/3} \) by \( y^{5/3} \)

Applying the distributive property is the first step in simplifying such expressions as it "breaks down" complex terms into simpler ones that can be handled separately.

Power of a Power Property

The power of a power property is another essential rule in exponentiation. This property states that when you multiply powers with the same base, you add their exponents. Mathematically, it’s expressed as \( a^m \times a^n = a^{m+n} \). Applying this rule simplifies expressions involving exponents.
For instance, using our problem, first apply this property to \( y^{1/3} \times y^{2/3} \). You simply add the exponents: \( \frac{1}{3} + \frac{2}{3} = 1 \). So, \( y^{1/3} \times y^{2/3} = y^1 = y \).
Next, use it for \( y^{1/3} \times y^{5/3} \): add the exponents \( \frac{1}{3} + \frac{5}{3} = 2 \). So \( y^{1/3} \times y^{5/3} = y^2 \). By using the power of a power property, we effectively reduce the complexity of operations with exponents.

Exponent Rules

Exponent rules, also known as laws of exponents, are guidelines that help simplify expressions involving powers or exponents. These rules include multiplying powers with the same base, dividing powers, and dealing with powers raised to another power. They are fundamental to manipulating algebraic expressions effectively.
In this exercise, we've seen exponent rules in action with multiplication:

• The rule \( a^m \times a^n = a^{m+n} \) was applied to combine terms.


• This allows simplification by reducing multiple factors with the same base into a single term with the summed exponent.

Mastering these rules opens the door to tackling a range of algebraic problems with confidence. With exponent rules, your toolkit in algebra becomes incredibly powerful, allowing you to simplify and solve complex equations with ease.