Question

Simplify each expression. \(\sqrt{11} \cdot \sqrt{11}\)

Short answer

The expression simplifies to 11.

Step-by-step solution

Recall the Property of Square Roots

When you multiply two square roots with the same value, they simplify to the number inside the root. Specifically, \( \sqrt{a} \cdot \sqrt{a} = a \). For our expression \( \sqrt{11} \cdot \sqrt{11} \), this property will be useful.

Apply the Property to the Expression

Apply the property from Step 1 to simplify the expression. By substituting the value, we have \( \sqrt{11} \cdot \sqrt{11} = 11 \).

Finalize the Simplification

The expression simplifies completely to 11, as we simplified it using the property of square roots that \( \sqrt{a} \cdot \sqrt{a} = a \). Thus, \( \sqrt{11} \cdot \sqrt{11} = 11 \).

Key concepts

Property of Square Roots

Understanding the property of square roots is essential when working with expressions involving roots. The property states that the product of two identical square roots is equal to the number inside the root. In other words, when you have an expression like \( \sqrt{a} \cdot \sqrt{a} \), it simplifies to \( a \). This is because taking a number’s square root and then squaring it are inverse operations, effectively canceling each other out.
For example, in our original exercise, we have \( \sqrt{11} \cdot \sqrt{11} \). Applying this property directly means that it simplifies to \( 11 \).

• This property helps to quickly find answers in multiplication of identical square roots.
• It is fundamental when simplifying more complex expressions involving square roots.

Remember, this property only applies when the numbers under the square roots are identical.

Simplifying Expressions

Simplifying expressions involving square roots can be straightforward if you understand the concepts behind it. When you see a square root expression like \( \sqrt{a} \) multiplied by itself, the goal is to simplify it to a more recognizable number. Using the property of square roots, as discussed earlier, makes this an easy task.
For example, let's look at \( \sqrt{11} \cdot \sqrt{11} \). The expression can immediately be rewritten using the property, resulting in \( 11 \). This approach shows how even complicated-looking expressions can be reduced to simpler forms using fundamental properties.

• Simplification aids in solving mathematical problems without lengthy calculations.
• It involves rewriting the expression in a simpler or more standardized form.

Always look for opportunities to apply known mathematical properties to simplify expressions.

Multiplication of Square Roots

The multiplication of square roots is done by multiplying the numbers inside the roots if they are the same, resulting in a whole number. Using our understanding of the property of square roots, this multiplication becomes very simple.
Consider, for example, \( \sqrt{11} \cdot \sqrt{11} \). Because both square roots contain the same number, multiplying them directly leads us to 11. This is because each square root effectively "undoes" the square on the number.

• If the values under each square root are different, you would multiply them together under a single square root.
• For identical square roots, the multiplication is a straightforward simplification using the property.

Being familiar with this process is crucial for both simplifying roots and performing efficient calculations in geometry and algebra.