Problem 63 - Algebra 1: Concepts and Skills

Question

Evaluate the radical expression when $a=-1 \text { and } b=5. $$\sqrt{b^{2}+9 a}$$

Step-by-Step Solution

Answer

The evaluated radical expression when \( a=-1 \) and \( b=5 \) is 4.

Step 1: Substitution

Substitute \( a \) with -1 and \( b \) with 5 into the expression \( \sqrt{b^{2}+9a} \), which becomes: \( \sqrt{5^{2}+9(-1)} \)

Step 2: Simplify the Expression under the Square Root

Solve the operation under the square root, \( 5^{2}+9(-1) = 25 - 9 = 16 \)

Step 3: Final Calculation

Calculate the square root of 16, which results in 4

Key Concepts

The Substitution Method

The substitution method is a straightforward way to solve expressions by replacing variables with known values. This method is useful when you have expressions where specific values for variables are provided. In practice,

• Identify the variables in the expression.
• Substitute each variable with the given numerical value.
• Perform arithmetic operations to simplify the expression further.

In our exercise, we were asked to find the value of the radical expression \( \sqrt{b^{2}+9a} \) with \( a = -1 \) and \( b = 5 \). By substituting \( a \) and \( b \) into the expression, we get \( \sqrt{5^{2}+9(-1)} \). This step lays the groundwork for simplification by turning an abstract expression into a concrete calculation.

Simplifying Expressions

Simplifying expressions is all about making calculations easier and more manageable. Once you've substituted values into an expression, the next step is to simplify it. This involves performing any necessary arithmetic operations.

• Start by dealing with exponents or powers, as they are higher priority operations.
• Follow through with multiplication and division.
• Finally, perform addition and subtraction.

In our specific example, we took \( 5^{2} \) to get 25. Then, we multiplied 9 by -1, resulting in -9. Subtracting the second result from the first gives us 16, which simplifies the expression under the square root to a simple, single number.

Understanding Square Roots

Square roots are fundamental in mathematics, especially when dealing with radical expressions. The square root operation finds a number which, when multiplied by itself, gives the original number under the radical. To solve a square root:

• Ensure the number inside the root, called the radicand, is non-negative to avoid complex numbers when working with real numbers.
• Look for numbers where the square root is known, typically perfect squares such as 1, 4, 9, 16, etc.

In our exercise, once the expression was simplified under the root, the radicand was 16. Since 16 is a perfect square \((4 \times 4)\), the square root operation is straightforward, resulting in an answer of 4. This step completes the evaluation of the radical expression, demonstrating the usefulness of knowing perfect squares for quick calculations.