Problem 43 - Algebra 1: Concepts and Skills

Question

Simplify the variable expression. $$ \left(y \cdot y^{1 / 3}\right)^{3 / 2} $$

Step-by-Step Solution

Answer

\( y^{2} \)

Step 1: Apply Power Rule

The power rule states that when raising a power to a power, you multiply the exponents together. Here \( y \) has 1 as implied exponent. So, apply the power rule \( (a^{m})^{n}=a^{mn} \) to get \( y^{1*3/2} \cdot y^{1/3*3/2} \).

Step 2: Expand the Exponents

Now, expand these exponents. That gives \( y^{3/2} \cdot y^{1/2} \).

Step 3: Apply Multiplication Rule of Exponents

The exponent rules reveal that when bases are the same, you add exponents together to multiply the expressions. Apply this multiplication rule to get \( y^{3/2+1/2} \).

Step 4: Wrap Up the Simplification

Now just wrap up the simplification and you'll have \( y^{2} \).

Key Concepts

Power Rule

The Power Rule is a fundamental concept in exponentiation that simplifies the process of dealing with exponents. When you have an expression where a power is raised to another power, like

• \((a^m)^n = a^{m \cdot n}\)

this rule tells you to multiply the exponents.
For example, in the expression provided,

• y is already raised to the 1st power as an implied exponent.
• So using the power rule
• \((a^{1})^{3/2} = a^{1\cdot 3/2} = a^{3/2}\).

This saves time and makes complex expressions easier to handle. Understanding this can hugely benefit solving any problems involving exponentiation quickly and effectively.
Next time you see nested exponents, remember the Power Rule to simplify your work.

Multiplication Rule of Exponents

Exponentiation can sometimes involve multiplying numbers with the same base, where the Multiplication Rule of Exponents is applied. This handy rule dictates that when you multiply like bases, you simply add their exponents together. This rule simplifies expressions to make them more manageable.
For instance, if we look again at our exercise step:

• You have two expressions, \( y^{3/2} \) and \( y^{1/2} \).
• Both share the base \( y \), so according to the rule you add the exponents:
• \( y^{3/2} \cdot y^{1/2} = y^{(3/2 + 1/2)} \).

By applying this rule, you can easily simplify a seemingly complex multiplication of variables involving exponents, saving time and reducing errors in computation.

Simplifying Expressions

Simplifying expressions is a crucial skill in mathematics, especially when working with exponents. Once you've used the Power Rule and the Multiplication Rule, you're often left with a simplified expression that needs one last step to reach a neat conclusion.
In our example, after applying the rules, you ended up with

• \( y^{3/2} \cdot y^{1/2} \), which simplifies to
• \( y^{2} \).

This means that all initially complex and large expressions are consolidated into a simpler form. This not only makes the expression easier to understand but also often reveals insights into the relationships between the elements of the expression. Simplifying expressions is not just about getting the right answer, but also about knowing the underlying mathematical principles.