Problem 64 - Prealgebra

Question

Compute the exact value of the given expression. \(-6 \sqrt{576}-8 \sqrt{121}\)

Step-by-Step Solution

Answer

The exact value of the expression is -232.

Step 1: Simplify the square root of 576

To simplify the expression, start by simplifying the square root of 576. The square root of 576 is calculated as follows: \( \sqrt{576} = 24 \). This is because 24 multiplied by 24 equals 576.

Step 2: Simplify the square root of 121

Next, simplify the square root of 121. The square root of 121 is \( \sqrt{121} = 11 \). This is because 11 multiplied by 11 equals 121.

Step 3: Substitute the simplified square roots

Now substitute the simplified values into the original expression: \(-6 \sqrt{576} - 8 \sqrt{121}\). This can be rewritten with the simplified values as \(-6 \times 24 - 8 \times 11\).

Step 4: Perform the multiplications

Calculate the products: \(-6 \times 24 = -144\) and \(-8 \times 11 = -88\).

Step 5: Add the results

Finally, add the two results from Step 4: \(-144 + (-88) = -232\).

Key Concepts

Understanding Square Roots

Square roots are a foundational topic in prealgebra, allowing us to work with numbers that are not perfect squares. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 576 is 24. This is because when you multiply 24 by 24, you get 576. Similarly, the square root of 121 is 11, because 11 times 11 equals 121.

When dealing with square roots, it's important to recognize both perfect squares and how to simplify the non-perfect ones. Perfect squares are numbers like 1, 4, 9, 16, and so on, which are squares of integers. Identifying perfect squares can greatly simplify calculations in exercises similar to the one we are solving. Once you determine the square root, you can replace the original root in the equation with this simpler term.

Simplification Steps

Simplification is key in mathematics as it makes expressions easier to manage. Whenever you encounter a complex expression, the first step is often to break it down into simpler parts. In our example, the expression involved two square roots that needed simplification:

• For \( \sqrt{576} \), we found it simplifies to 24.
• For \( \sqrt{121} \), it simplifies to 11.

Simplifying these terms reduces the complex expression into something more manageable and easier to work with.

Furthermore, the approach doesn't just apply to square roots. You should apply simplification to any calculable parts of your expressions for ease of solving. It's about making your operations efficient and ensuring you have the simplest form of your equations before performing further calculations.

Performing Arithmetic Operations

Arithmetic operations involve basic math functions such as addition, subtraction, multiplication, and division. In our exercise, we focused on multiplication and subtraction. After the square roots were simplified, we had

• \(-6 \times 24\)
• \(-8 \times 11\)

Both are multiplication operations. We multiply the coefficients by the simplified square roots. In arithmetic, particularly multiplication in this context, negative signs are crucial. It's essential to account for them to maintain the correct sign of the outcome.

After performing these multiplications:

• \(-6 \times 24\) resulted in \(-144\)
• \(-8 \times 11\) resulted in \(-88\)

Finally, adding these terms is straightforward if the operations are error-free: \(-144 + (-88) = -232\). This exercise showcases the importance of performing arithmetic operations accurately to reach the correct total.