Problem 100 - Algebra and Trigonometry

Question

Simplify using properties of exponents. $$ \frac{\left(2 y^{\frac{1}{5}}\right)^{4}}{y^{\frac{3}{10}}} $$

Step-by-Step Solution

Answer

The simplified form of the given expression \(\frac{\left(2 y^{\frac{1}{5}}\right)^{4}}{y^{\frac{3}{10}}}\) is \(16y^{\frac{1}{2}}\)

Step 1: Applying the power of a product rule

The exercise gives \(\left(2 y^{\frac{1}{5}}\right)^{4}\). Using the power of a product rule, this simplifies to \(2^{4} * y^{\frac{4}{5}}\) which is equal to \(16y^{\frac{4}{5}}\)

Step 2: Division of Like Bases Exponents

Now, divide this term by \(y^{\frac{3}{10}}\). Going by the division of same base exponent rule, we subtract their exponent values . Hence the term becomes \(16y^{\frac{4}{5}-\frac{3}{10}}\) or \(16y^{\frac{8}{10}-\frac{3}{10}}\)

Step 3: Simplifying the expression

After subtraction, the expression simplifies to \(16y^{\frac{5}{10}}\). As the exponent \(\frac{1}{2}\) is equivalent to \(\frac{5}{10}\), simplify this expression further to \(16y^{\frac{1}{2}}\)

Key Concepts

Power of a Product Rule

When faced with an expression where a product is raised to a power, the power of a product rule comes in handy. Essentially, this rule allows us to distribute the exponent across each of the factors inside the parentheses. For the exercise \((2 y^{\frac{1}{5}})^{4}\), you apply the rule by raising both 2 and \(y^{\frac{1}{5}}\) to the 4th power. This transforms the expression into \(2^{4} \cdot (y^{\frac{1}{5}})^{4}\).
Breaking it down further:

• \(2^{4}\) becomes 16, because 2 multiplied by itself four times equals 16.
• \((y^{\frac{1}{5}})^{4}\) is simplified to \(y^{\frac{4}{5}}\) by multiplying the exponents, i.e., \(\frac{1}{5} \times 4\). This results in \(y^{\frac{4}{5}}\).

Putting it all back together gives us \(16y^{\frac{4}{5}}\). This rule is a powerful tool in simplifying expressions by reducing the complexity of brackets.

Division of Exponents

When you divide expressions with exponents that have the same base, the division of exponents rule is applied. This rule states that you subtract the exponent of the denominator from the exponent of the numerator if the bases are the same.
For example, taking \(\frac{y^{\frac{4}{5}}}{y^{\frac{3}{10}}}\), where both terms have the base \(y\), you subtract \(\frac{3}{10}\) from \(\frac{4}{5}\). Adjust the fractions to have a common denominator to make the subtraction straightforward.

• Convert \(\frac{4}{5}\) into \(\frac{8}{10}\) so that it matches the denominator of \(\frac{3}{10}\).
• Subtracting these, \(\frac{8}{10} - \frac{3}{10}\), results in \(\frac{5}{10}\).

Keep in mind that when using this rule, it only applies to expressions where the bases are the same.

Simplifying Expressions

Once you’ve used the division of exponents rule, you will need to simplify the resulting expression. After performing the division in the example, you’re left with \(16y^{\frac{5}{10}}\).
Since \(\frac{5}{10}\) can be simplified further to \(\frac{1}{2}\), the expression becomes \(16y^{\frac{1}{2}}\). Here, simplifying exponents is just like simplifying fractions.
Reducing expressions to their simplest form often involves identifying fractions or numbers that can be broken down into smaller, equivalent parts. This step ensures that the expression is in its most manageable form.

Like Bases Exponents

Working with like bases is foundational when simplifying algebraic expressions involving exponents. Like bases refer to terms in which the variable part is the same, such as the \(y\) in \(y^{\frac{4}{5}}\) and \(y^{\frac{3}{10}}\). When the bases are the same, operations such as multiplication and division become much simpler as exponents can be directly manipulated.
Operations with like bases often involve:

• Addition of exponents when multiplying.
• Subtraction of exponents when dividing.

It's important to always watch for like bases in an expression as they allow for the direct application of exponent properties, making the simplification process both efficient and systematic.