Problem 31 - Elementary Algebra

Question

Find each of the following products. $$ \sqrt{y} \sqrt{y} $$

Step-by-Step Solution

Answer

Answer: The product of the square roots $\sqrt{y}$ and $\sqrt{y}$ is $y$.

Step 1: Multiply the square roots

To start, we need to multiply the two square roots: $$ \sqrt{y} \times \sqrt{y} $$

Step 2: Apply the rule for multiplying square roots

Now, we apply the rule for multiplying square roots of the same radicand: $$ \sqrt{a}\times \sqrt{a} = a $$ In our case, the radicand a is y. So we have: $$ \sqrt{y} \times \sqrt{y} = y $$

Step 3: Simplify and write the answer

The result has already been simplified: $$ \sqrt{y} \sqrt{y} = y $$

Key Concepts

Simplifying Radicals

Simplifying radicals is all about making the expression under the square root as simple as possible. It's like cleaning up your room by putting away extra clutter. When \(\sqrt{a} \times \sqrt{a}\), you're essentially combining two identical groups under the same root, which simplifies to just \(a\).
In our example, multiplying \(\sqrt{y} \times \sqrt{y}\) simplifies directly to \(y\). When you see repeated radicals like this, the operation is straightforward and results in the value under the root. For those still learning, remember this handy fact:
\[\sqrt{a} \times \sqrt{a} = a\] which cleans up the entire expression into a simple non-radical form. When simplifying radicals, always check if the values under the square root are perfect squares as this immediately simplifies the entire operation.

Algebraic Expressions

Algebraic expressions often involve a mix of numbers, variables, and operations. They can be as simple as \(x + 3\) or involve more complex operations like \(\sqrt{y}\). When dealing with expressions that contain radicals, such as \(\sqrt{y}\), it's important to treat the radical as an algebraic term in its own right.
Variables like \(y\) can stand for any number, which make expressions both versatile and occasionally tricky. If \(y\) is a perfect square, then \(\sqrt{y}\) becomes a simple, non-radical number. In the expression \(\sqrt{y} \times \sqrt{y}\), you're multiplying two identical radicals, which simplifies to \(y\), a crucial step in algebraic calculations.

• Radicals behave similarly to variable terms in multiplication.
• Ensure you're applying the correct arithmetic rule, such as multiplying similar terms or simplifying multiples.

Concentration is key—while they look complicated, algebraic expressions help us solve real-world problems with ease.

Properties of Radicals

Radicals, especially square roots, have some handy properties which make them easier to work with. These properties help us manipulate and simplify expressions involving roots. For the problem \(\sqrt{y}\times\sqrt{y} = y\), understanding these basic properties is crucial:
**Multiplication Property:** When you multiply square roots with the same radicand, it simplifies to the non-root form. This means \(\sqrt{a} \times \sqrt{a} = a\), a key property used here.
**Product Property:** You can multiply the numbers inside the radicals, a useful tactic for cases where you don’t have identical radicands. For example, \(\sqrt{x}\times\sqrt{y} = \sqrt{xy}\).
**Simply Multiply:** If radicals share the same index and radicand, multiply the radicands directly and reconsider the root once simplified.
These properties guide us in simplifying and understanding expressions, ensuring we work with them efficiently and accurately. Always keep these rules in your toolkit when working with radicals to handle any expression confidently.