Problem 66 - Algebra 2

Question

Simplify each expression. Assume that all variables are positive. $$y^{\frac{1}{2}} \cdot y^{\frac{3}{10}}$$

Step-by-Step Solution

Answer

\(y^{\frac{4}{5}}\)

Step 1: Identify the base and exponents

In \(y^{\frac{1}{2}} \cdot y^{\frac{3}{10}}\), 'y' is the base and \(\frac{1}{2}\) and \(\frac{3}{10}\) are the exponents.

Step 2: Apply the rule of exponents

The rule of exponents states that when multiplying terms with the same base, we add the exponents. Therefore, we add up \(\frac{1}{2}\) and \(\frac{3}{10}\).

Step 3: Convert the mixed fractions into an improper fraction

We convert the fraction \(\frac{1}{2}\) into 10 as the denominator to add them up easily. So, \(\frac{1}{2}\) is \(\frac{5}{10}\). Thus, \(y^{\frac{5}{10}} \cdot y^{\frac{3}{10}}\)

Step 4: Add the fractional exponents

Adding up \(\frac{5}{10}\) and \(\frac{3}{10}\) we get \(\frac{8}{10}\). So, \(y^{\frac{5}{10}} \cdot y^{\frac{3}{10}} = y^{\frac{8}{10}}\)

Step 5: Simplify the exponents

We can simplify the fraction \(\frac{8}{10}\) to \(\frac{4}{5}\). Therefore, our final answer is \(y^{\frac{4}{5}}\)

Key Concepts

Simplifying Expressions

Simplifying mathematical expressions is an essential part of algebra. It involves rewriting expressions in a form that is easier to understand or work with. One might simplify an expression to make calculations clearer or more straightforward to solve.
Begin by identifying components like variables, coefficients, and any operations such as addition, subtraction, multiplication, or division. Ask yourself: Can these operations be performed more directly or combined using mathematical rules?
For instance, consider simplifying the expression \(y^{\frac{1}{2}} \cdot y^{\frac{3}{10}}\). Here, the variable \(y\) appears in both terms, which signals that the expression can potentially be compacted.

• Look for elements with common bases or similar structures.
• Apply relevant mathematical laws, such as laws of exponents, to simplify.
• Aim for the simplest form, where the expression retains its meaning while being less complex.

Simplification not only speeds up problem-solving but also provides better insights into the variables involved in an equation.

Fractional Exponents

Fractional exponents are a way to express roots and powers in one notation. They can look intimidating at first, but they're just another way of dealing with roots.
The denominator of a fractional exponent indicates the root, while the numerator indicates the power. For example, \(y^{\frac{1}{2}}\) is equivalent to the square root of \(y\), whereas \(y^{\frac{3}{10}}\) suggests taking the 10th root of \(y\) and raising it to the 3rd power.
Understanding these exponents can greatly simplify calculations, especially when dealing with complex polynomials or algebraic equations.

• Remember that \(a^{\frac{m}{n}} = \sqrt[n]{a^m}\).
• Fractional exponents allow for easier manipulation of terms without needing to switch between powers and roots.
• The flexibility of fractional notation lets you handle logarithmic and exponential relationships more elegantly.

By becoming comfortable with fractional exponents, you can tackle a wider range of problems with confidence.

Rules of Exponents

The rules of exponents are vital tools in algebra that help in simplifying expressions, particularly those involving powers and roots. These rules communicate how to efficiently handle expressions with exponents.
When dealing with the same base, particularly in multiplication, the primary rule is simple: add the exponents. For example, in the expression \(y^{\frac{1}{2}} \cdot y^{\frac{3}{10}}\), both terms have a base of \(y\). Thus, according to the rules, their exponents can be combined through addition.
Let's outline the key rules you’ll frequently use:

• A product rule: \(x^m \cdot x^n = x^{m+n}\).
• A quotient rule: \(\frac{x^m}{x^n} = x^{m-n}\), assuming \(x eq 0\).
• A power of a power rule: \((x^m)^n = x^{m \cdot n}\).

Applying these rules correctly not only simplifies calculations but also helps unravel complex expressions into manageable bits. Above all, these rules bridge the gap between numerical and algebraic thinking.