Problem 89 - Intermediate Algebra

Question

Perform the multiplications. All variables represent positive real numbers. See Example \(9 .\) $$ y^{1 / 3}\left(y^{2 / 3}+y^{5 / 3}\right) $$

Step-by-Step Solution

Answer

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

Step 1: Distribute the term

We will distribute the term outside of the parentheses across each term within the parentheses. This follows the distributive property: \[a(b + c) = ab + ac\] In this case, distribute \(y^{1/3}\) across \(y^{2/3}\) and \(y^{5/3}\):\[y^{1/3} imes y^{2/3} + y^{1/3} imes y^{5/3}\]

Step 2: Apply exponent rules

We will now multiply the terms by adding their exponents, as both terms share the base \(y\). The rule for multiplying exponents with the same base is:\[a^m imes a^n = a^{m+n}\]First term:\[y^{1/3} imes y^{2/3} = y^{(1/3) + (2/3)} = y^{3/3} = y^1 = y\]Second term:\[y^{1/3} imes y^{5/3} = y^{(1/3) + (5/3)} = y^{6/3} = y^2\]

Step 3: Write the final expression

Now that we've multiplied the terms, we can write the final expression as the sum of the two simplified products:\[y + y^2\]

Key Concepts

Exponent Rules

Exponent rules help us navigate mathematical expressions that involve powers. An exponent tells us how many times to multiply a number by itself. For instance, the expression \(a^n\) means we multiply 'a' by itself 'n' times.

When dealing with multiplication of terms with the same base, there is a specific rule to follow. This is known as the rule of multiplying powers. In this case, if you have two terms like \(a^m\) and \(a^n\), the result is \(a^{m+n}\). This means you add the exponents together while keeping the base the same.

• For example, \(x^2 \times x^3 = x^{2+3} = x^5\).
• This same rule applies to divide terms where you subtract exponents instead of add, i.e., \(a^m / a^n = a^{m-n}\).

Understanding these rules simplifies expressions and makes algebraic operations more manageable.

Positive Real Numbers

Positive real numbers are numbers greater than zero. They include all the numbers on the number line to the right of zero, excluding zero itself. These numbers can be whole numbers, fractions, or decimals that are not negative.

In the context of algebra, positive real numbers are significant because they ensure that expressions involving variables result in discernible real number outputs. This is particularly important in functions and equations where such numbers are used as inputs or coefficients.

Here are some simple characteristics of positive real numbers:

• They are always greater than zero.
• They can be represented as fractions, such as \(2/3\) or as decimals like \(3.14\).
• They are often used in algebra to represent variable quantities that have no sign restrictions.

Understanding the inherent properties of positive real numbers allows mathematicians and students alike to predict the behavior of algebraic expressions and solve equations accurately.

Multiplication in Algebra

Multiplication is a foundational operation in algebra, and it involves finding the product of numbers or expressions. The process becomes slightly different when dealing with variables and algebraic terms.

A key principle in algebraic multiplication is the distributive property, which allows us to multiply a term across a sum or difference. This property is expressed as \(a(b + c) = ab + ac\). By applying this property, you can simplify expressions and streamline solving equations.

• If you have \(y^{1/3}(y^{2/3} + y^{5/3})\), distribute to get \(y^{1/3} \cdot y^{2/3} + y^{1/3} \cdot y^{5/3}\).
• After distribution, using the exponent rules helps to combine terms efficiently.

By mastering multiplication and the distributive property in algebra, you enable yourself to solve more complex problems systematically.