Problem 61 - Intermediate Algebra

Question

Multiply. $$ y^{1 / 2}\left(y^{1 / 2}-y^{2 / 3}\right) $$

Step-by-Step Solution

Answer

y - y^{7/6}

Step 1: Distribute the Multiplication

We need to multiply each term inside the parentheses by the term outside the parentheses. We start by distributing: \( y^{1/2} \times y^{1/2} \) and \( y^{1/2} \times -y^{2/3} \).

Step 2: Apply the Exponent Rule for Multiplication

Recall that when you multiply terms with the same base, you add their exponents. So, \( y^{1/2} \times y^{1/2} = y^{(1/2)+(1/2)} = y^{1} = y \).

Step 3: Simplify the Second Term

Now, apply the same rule to the second term:\( y^{1/2} \times -y^{2/3} = -y^{(1/2)+(2/3)} \).First, find a common denominator for the exponents, which is 6:\( \frac{1}{2} + \frac{2}{3} = \frac{3}{6} + \frac{4}{6} = \frac{7}{6}. \)Therefore, the product of the second term is:\( -y^{7/6} \).

Step 4: Combine the Results

Now combine the results from Steps 2 and 3 to give the final expression:\( y^{1/2}(y^{1/2} - y^{2/3}) = y - y^{7/6}.\)

Key Concepts

Distribution in Algebraic Expressions

Distribution is a method used when multiplying a single term by each of the terms inside a set of parentheses. It follows the distributive law, which states:

• For an expression like \( a(b + c) \), you multiply \( a \) by both \( b \) and \( c \) separately.

This often helps in breaking down complex problems into simpler parts. In the given exercise, distribution is applied by multiplying \( y^{1/2} \) with each term inside the parentheses, specifically with \( y^{1/2} \) and \(-y^{2/3} \).
By distributing, you effectively simplify the expression and prepare it for further operations like applying the exponent rules.
Distribution ensures that each term is correctly multiplied, which is critical in obtaining an accurate result.

Understanding Exponent Rules

Exponent rules are essential when dealing with expressions that include powers, like those seen in the exercise. Learning these rules can simplify your calculations:

• The product rule states that when you multiply two powers with the same base, you can add their exponents: \( a^m \times a^n = a^{m+n} \).
• In the exercise, this rule helps to compute \( y^{1/2} \times y^{1/2} = y^{1} \).
• When multiplying \( y^{1/2} \times -y^{2/3} \), you also add the exponents: \(\frac{1}{2} + \frac{2}{3} \).

By rewriting 1/2 and 2/3 with a common denominator of 6, you get \(\frac{3}{6} + \frac{4}{6} = \frac{7}{6} \), leading to \(-y^{7/6} \).
Using exponent rules helps make any algebraic manipulation involving powers straightforward and does away with the complex manual expansion of powers.

Simplifying Algebraic Expressions

Algebraic expressions are collections of numbers, variables, and operation signs. Simplifying them means reducing the expression to its most concise form without changing its value. In our context, after applying the distribution and exponent rules, you combine terms to arrive at a simpler, more unified expression.

• After applying the distribution, you have gotten two separate terms: \( y \) and \( -y^{7/6} \).
• The final step is simply to write these terms together, resulting in \( y - y^{7/6} \).

This combines the results into a cohesive expression that represents the operation performed, often a crucial last step in algebraic simplification tasks. By mastering this process, you develop an ability to efficiently handle more complex equations in future problems.