Problem 69 - Algebra and Trigonometry

Question

\(69-82\) . Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers. $$ \left(\sqrt[6]{y^{5}}\right)\left(\sqrt[3]{y^{2}}\right) $$

Step-by-Step Solution

Answer

The expression simplifies to \( y^{\frac{3}{2}} \).

Step 1: Express Radicals as Exponents

Convert the radicals to exponential form.\( \sqrt[6]{y^{5}} \) can be rewritten as \( y^{\frac{5}{6}} \).\( \sqrt[3]{y^{2}} \) can be rewritten as \( y^{\frac{2}{3}} \).

Step 2: Multiply the Exponential Terms

When multiplying terms with the same base, you add the exponents.The expression is now \( y^{\frac{5}{6}} \cdot y^{\frac{2}{3}} \).Add the exponents: \( \frac{5}{6} + \frac{2}{3} \).

Step 3: Simplify the Exponents

Find a common denominator to add \( \frac{5}{6} + \frac{2}{3} \).Convert \( \frac{2}{3} \) to \( \frac{4}{6} \) so that the common denominator is 6.Add: \( \frac{5}{6} + \frac{4}{6} = \frac{9}{6} \), which simplifies to \( \frac{3}{2} \).

Step 4: Express the Final Answer

Write the expression with the simplified exponent:The expression becomes \( y^{\frac{3}{2}} \).

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential skill in algebra. It involves reducing an expression to its most basic form without changing its value. In the context of rational exponents, this often means converting complex expressions into a simpler form.
One common method of simplifying is combining like terms, where terms with the same variable and exponent can be added or subtracted. Additionally, when expressions with exponents are involved, using the property of exponents can greatly reduce complexity.
For expressions involving multiplication of terms with the same base, remember:

• Add the exponents together since multiplying adds their powers.
• Simplify the resulting exponent if possible.

For instance, given the expression \[ y^{\frac{5}{6}} \times y^{\frac{2}{3}}, \] we combine the exponents by finding a common denominator. Once combined and simplified, the expression is \[ y^{\frac{3}{2}}. \] This is a simplified form, where the complex roots and powers are expressed with rational exponents.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. These can sometimes look challenging but can be simplified with the use of rational exponents.
To transform a radical expression into an expression with a rational exponent, use the principle that the root is the denominator and the power inside the root is the numerator. For example, the expression \( \sqrt[6]{y^{5}} \) is equivalent to \( y^{\frac{5}{6}}. \)
Simplifying radical expressions helps you to work more easily with the expressions and solve equations. This process is crucial when solving problems because it converts difficult-to-manage radicals into manageable exponential terms.

Exponent Rules

Exponent rules are key concepts that allow us to manipulate and simplify mathematical expressions involving powers. Here are some fundamental rules:

• Product of Powers Rule: When multiplying like bases, add the exponents, i.e., \( a^m \times a^n = a^{m+n}. \)
• Power of a Power Rule: When raising an exponent to another power, multiply the exponents, i.e., \( (a^m)^n = a^{m \times n}. \)
• Quotient of Powers Rule: When dividing like bases, subtract the exponents, i.e., \( \frac{a^m}{a^n} = a^{m-n}. \)

In our example, we used the product of powers rule to simplify \( y^{\frac{5}{6}} \times y^{\frac{2}{3}} \) by adding the exponents once they shared a common denominator. Remember, understanding exponent rules paves the way for working easily with more complicated algebraic expressions and is fundamental in algebra and calculus.