Problem 4 - Algebra for College Students

Question

Evaluate each of the following. For example, \(\sqrt{25}=5\). \(-\sqrt{81}\)

Step-by-Step Solution

Answer

The solution to \\(-\sqrt{81}\\) is \\(-9\\).

Step 1: Simplify the Square Root

To solve \(-\sqrt{81}\), first find the positive square root of 81. Calculate \( \sqrt{81} = 9 \).This is because \(9 \times 9 = 81\).

Step 2: Apply the Negative Sign

Once you have the square root value, apply the negative sign to the result to get the complete solution.Thus, \(-\sqrt{81} = -9\).

Key Concepts

Simplifying Square Roots

When we talk about simplifying square roots, we mean finding the base number that, when multiplied by itself, equals the number under the square root sign. For instance, if we have \( \sqrt{81} \), we are searching for a number \( x \) such that \( x^2 = 81 \). Here, \( x = 9 \) because \( 9 \times 9 = 81 \). This process is straightforward for perfect squares, numbers like 1, 4, 9, 16, 25, and so on, that can be expressed as the square of integers. For these numbers, you can quickly identify the square root by using multiplication facts or a multiplication table.

• Identify if the number is a perfect square by checking if a whole number multiplied by itself gives the number under the root.
• Example: For \( \sqrt{16} \), since \( 4 \times 4 = 16 \), the square root is 4.

Knowing the perfect squares up to at least \( 144 \), which is \( 12 \times 12 \), gives you a handy reference for simplifying most basic square root expressions. If the number isn't a perfect square, further techniques or approximations might be needed, but our current scope is just on perfect squares.

Negative Square Roots

The concept of negative square roots might sound intimidating, but it essentially involves a simple twist. When we see an expression like \(-\sqrt{81}\), we're dealing with the negative of the square root of 81.First, solve the square root as if the negative sign isn't there. You find \( \sqrt{81} = 9 \), because \( 9 \times 9 = 81 \). The negative part simply means you then multiply the result by \(-1\), yielding \(-9\).

• The minus sign means the result is the opposite of the positive square root.
• Example: \(-\sqrt{49} = -7\) because the square root of 49 is 7, and then you apply the negative.

It's important to distinguish between the square root of a number and the negative square root. The first, \( \sqrt{81} \), is 9, whereas \(-\sqrt{81} \) results in \(-9\). This is simple but crucial to solving these types of problems correctly.

Step-by-Step Solutions

Step-by-step solutions are essential for understanding how to systematically tackle problems, especially with square roots. Let's break down the solution for \(-\sqrt{81} \) into easy steps.**Step 1: Simplify the Square Root**First, calculate the square root of 81. Since \( 9 \times 9 = 81 \), we find that \( \sqrt{81} = 9 \). Knowing this basic math fact quickly helps simplify our initial problem.**Step 2: Apply the Negative Sign**Once you have the positive square root, apply the negative sign to it. This means converting \( 9 \) into \(-9\), which gives us the final answer: \(-\sqrt{81} = -9\).

• Start by isolating the basic square root calculation.
• Maintain the order by evaluating the square root first before applying any additional operations such as negative signs.

By unfolding problems step-by-step, you gain a clearer sense of not just what the answer is, but more importantly, why it is what it is. It also instills confidence in handling both simple and more complex mathematical challenges.