Problem 44 - Algebra A Combined Function

Question

Add or subtract as indicated. Assume that all variables represent positive real numbers. $$ \frac{\sqrt[3]{y^{5}}}{8}+\frac{5 y \sqrt[3]{y^{2}}}{4} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{11y^{5/3}}{8} \).

Step 1: Simplifying the First Term

The first term is \( \frac{\sqrt[3]{y^{5}}}{8} \). We can rewrite this by recognizing that \( \sqrt[3]{y^5} = y^{5/3} \). Therefore, the term becomes \( \frac{y^{5/3}}{8} \).

Step 2: Simplifying the Second Term

The second term is \( \frac{5y \sqrt[3]{y^2}}{4} \). Again, using the cube root rule, \( \sqrt[3]{y^2} = y^{2/3} \). So this term becomes \( \frac{5y \cdot y^{2/3}}{4} = \frac{5y^{1+2/3}}{4} = \frac{5y^{5/3}}{4} \).

Step 3: Finding a Common Denominator

We need to add \( \frac{y^{5/3}}{8} \) and \( \frac{5y^{5/3}}{4} \). The common denominator of 8 and 4 is 8. Convert \( \frac{5y^{5/3}}{4} \) to have a denominator of 8: \( \frac{5y^{5/3} \times 2}{4 \times 2} = \frac{10y^{5/3}}{8} \).

Step 4: Adding the Two Terms

Now that both terms have a common denominator, we can add them: \( \frac{y^{5/3}}{8} + \frac{10y^{5/3}}{8} = \frac{y^{5/3} + 10y^{5/3}}{8} = \frac{11y^{5/3}}{8} \).

Step 5: Final Simplified Expression

The final expression is \( \frac{11y^{5/3}}{8} \), which is the simplified form of the given expression.

Key Concepts

Cube Roots

When dealing with cube roots, we're asking what number, when multiplied by itself three times, gives us the original number inside the radical. Unlike square roots, which look for pairs, cube roots are all about triples. For instance, the cube root of 8 is 2 because \( 2 \times 2 \times 2 = 8 \). In our exercise, we see cube roots in expressions like \( \sqrt[3]{y^5} \) and \( \sqrt[3]{y^2} \). Here, it's important to translate these cube roots into expressions that are easier to work with.

• For \( \sqrt[3]{y^5} \), think of it as \( y^{5/3} \). You are using the property \( \sqrt[3]{y^n} = y^{n/3} \).
• Similarly, \( \sqrt[3]{y^2} \) transforms to \( y^{2/3} \). This makes addition and multiplication much simpler.

Getting comfortable with cube roots and knowing how to convert them to fractional exponents will help immensely in working through algebraic expressions.

Simplifying Expressions

Simplifying expressions is all about reducing complexity. This makes them easier to work with or understand. When you simplify, you're essentially making sure that you express the components in their most reduced form. In our example, we simplified the expression \( \frac{5y \sqrt[3]{y^2}}{4} \).

• First, convert cube roots to fractional exponents, \( y^{2/3} \), then multiply by the existing variable \( y \).
• Using the rule \( y^a \times y^b = y^{a+b} \), combine these terms into \( y^{1+2/3} = y^{5/3} \).
• Rewriting terms in this way ensures all terms are expressed similarly, ready for further operation like addition or subtraction.

By identifying parts of the expression that can be modified like this, you gain an upper hand in dealing with long and tricky equations.

Common Denominator

Combining fractions with different denominators requires finding a common denominator. This step ensures that you can add or subtract fractions easily. Let's dive into our exercise. We derived two expressions: \( \frac{y^{5/3}}{8} \) and \( \frac{5y^{5/3}}{4} \). Here's how you find a common denominator:

• Identify the least common multiple (LCM) of the denominators. For 8 and 4, the LCM is 8.
• Convert each fraction so they both have the new denominator. Keep the value unchanged by multiplying numerator and denominator accordingly.
• The second term \( \frac{5y^{5/3}}{4} \) becomes \( \frac{10y^{5/3}}{8} \) by multiplying top and bottom by 2. Now they can be easily added.

Finding a common denominator is a crucial skill in algebra. Once both fractions share the same base, you can combine them into one, thus simplifying the overall problem-solving process.