Problem 2 - Algebra 2 and Trigonometry

Question

Does \(\sqrt{16}+\sqrt{48}=\sqrt{64} ?\) Justify your answer.

Step-by-Step Solution

Answer

No, because \(4 + 4\sqrt{3} \neq 8\).

Step 1: Simplify each square root

We start by simplifying each square root in the equation. First, simplify \(\sqrt{16}\): \(\sqrt{16} = 4\), because \(4^2 = 16\). Next, simplify \(\sqrt{48}\): \(\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3}\). Finally, \(\sqrt{64} = 8\) because \(8^2 = 64\).

Step 2: Evaluate the left side of the equation

Add the simplified terms on the left side. We have \(\sqrt{16} + \sqrt{48} = 4 + 4\sqrt{3}\).

Step 3: Compare both sides of the equation

The left side of the equation is \(4 + 4\sqrt{3}\), and the right side is \(8\). Since \(4 + 4\sqrt{3}\) does not equal \(8\), the equation \(\sqrt{16} + \sqrt{48} = \sqrt{64}\) is not valid.

Key Concepts

Simplifying Radicals

Simplifying radicals is an essential part of working with square roots. A radical expression involves the square root symbol and the number inside it, called the radicand. Simplifying radicals means breaking down these expressions into their simplest form.

• To simplify \(\sqrt{16}\), recognize that 16 is a perfect square and equals \(4^2\). Thus, \(\sqrt{16} = 4\).
• For numbers that aren’t perfect squares, such as 48, find the largest perfect square factor. In this case, \(16 \text{ is a factor of } 48\), leading to \(\sqrt{48} = \sqrt{16 \times 3}\). Then, \(\sqrt{48} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3}\).
• Lastly, \(\sqrt{64}\) simplifies to 8, as \(8^2 = 64\).

Understanding this process allows you to handle more complex mathematical problems by working with simpler, easier-to-understand numbers.

Equation Solving

Equation solving involves finding values that make the equation true. In our case, we evaluated whether the equation \(\sqrt{16}+\sqrt{48}=\sqrt{64}\) holds true.

• First, simplify both sides of the equation by breaking down each radical into its simplest form. This helps us see the realistic values each side represents.
• The left side simplifies to \(4 + 4\sqrt{3}\), while the right side becomes \(8\). To solve the equation, we compare these values.
• Needless to say, \(4 + 4\sqrt{3}\) and \(8\) are not equal, which shows the original equation is not valid.

Approaching equations in this step-by-step manner helps in understanding discrepancies and resolving mathematical expressions accurately.

Inequality

Inequalities involve expressions that are not equal but rather have a relationship defined by greater than, less than, or similar terms. After simplifying the radicals in the given equation, we compared two expressions: \(4 + 4\sqrt{3}\) and \(8\).

• The left side of the equation simplifies to \(4 + 4\sqrt{3}\) while the right side provides us with \(8\).
• When comparing these, we realize that \(4 + 4\sqrt{3} > 8\) since \(4\sqrt{3}\) contributes more than \(4\) on its own.
• This indicates a type of inequality rather than an equation, highlighting that two sides do not equal but differ significantly.

Understanding inequalities is important in various aspects of mathematics as they describe the range of possible solutions, offering insights into the relationships among different mathematical expressions.