Simplifying and Subtracting Square Roots

The problem asks us to find the value of the expression $\sqrt{150}-\sqrt{54}-\sqrt{24}$. To solve this, we need to simplify each square root term by factoring out perfect squares.

Step 1: Simplify Each Square Root Term

We will simplify each term separately:

• Simplifying √150: Find the largest perfect square factor of 150. We know that $150 = 25 \times 6$. Since 25 is a perfect square ($5^2$), we can rewrite $\sqrt{150}$ as $\sqrt{25 \times 6}$. Using the property $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$, we get $\sqrt{25} \times \sqrt{6} = 5\sqrt{6}$.
• Simplifying √54: Find the largest perfect square factor of 54. We know that $54 = 9 \times 6$. Since 9 is a perfect square ($3^2$), we can rewrite $\sqrt{54}$ as $\sqrt{9 \times 6}$. This gives us $\sqrt{9} \times \sqrt{6} = 3\sqrt{6}$.
• Simplifying √24: Find the largest perfect square factor of 24. We know that $24 = 4 \times 6$. Since 4 is a perfect square ($2^2$), we can rewrite $\sqrt{24}$ as $\sqrt{4 \times 6}$. This results in $\sqrt{4} \times \sqrt{6} = 2\sqrt{6}$.

Step 2: Substitute Simplified Terms into the Expression

Now substitute the simplified forms back into the original expression:

$\sqrt{150}-\sqrt{54}-\sqrt{24} = (5\sqrt{6}) - (3\sqrt{6}) - (2\sqrt{6})$

Step 3: Combine Like Terms

All the terms now have the same radical part, which is $\sqrt{6}$. These are called like terms, similar to how $5x$, $3x$, and $2x$ are like terms. We can combine them by performing the subtraction on their coefficients:

$(5 - 3 - 2)\sqrt{6}$

Step 4: Perform the Subtraction of Coefficients

Calculate the value inside the parentheses:

$5 - 3 = 2$

$2 - 2 = 0$

So, the expression becomes:

$(0)\sqrt{6}$

Step 5: Final Result

Any number multiplied by 0 is 0. Therefore:

$(0)\sqrt{6} = 0$

The value of the expression $\sqrt{150}-\sqrt{54}-\sqrt{24}$ is 0.

Revision Table: Square Root Simplification

Additional Information: Understanding Radical Expressions

A radical expression is an expression that contains a square root (or cube root, etc.). Simplifying radical expressions often involves factoring the number under the radical sign (the radicand) to find perfect square factors. The property $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ allows us to pull the square root of the perfect square out of the radical. For example, $\sqrt{36} = \sqrt{4 \times 9} = \sqrt{4} \times \sqrt{9} = 2 \times 3 = 6$, which is correct since $\sqrt{36}=6$. When adding or subtracting radical expressions, you can only combine 'like' radicals, which are terms that have the exact same radical part (e.g., $5\sqrt{6}$ and $3\sqrt{6}$). Think of $\sqrt{6}$ as a variable; you combine the coefficients just like with algebraic terms.