Problem 69 - Algebra 1: Concepts and Skills

Question

Evaluate the radical expression when $a=-1 \text { and } b=5. $$\frac{\sqrt{86+a b}}{a}$$

Step-by-Step Solution

Answer

The evaluated value of the given radical expression is -9

Step 1: Substitution of the Given Values

Firstly, substitute the given values of 'a' and 'b' into the expression. So, replace 'a' with -1 and 'b' with 5. This will give a new expression: \(\frac{ \sqrt{86+(-1)(5)}}{-1}\)

Step 2: Simplify under the Root

Now, simplify the expression under the square root by performing the multiplication. This will give the modified expression: \(\frac{\sqrt{86 - 5}}{-1}\) After simplification, the expression reduces to: \(\frac{\sqrt{81}}{-1}\)

Step 3: Evaluate the Root

Evaluate the square root of 81, which yields 9. Therefore, the expression becomes: \(\frac{9}{-1}\)

Step 4: Perform the Division

To complete the evaluation, perform the division of 9 by -1, which gives -9. Thus, the expression evaluates to -9

Key Concepts

Substitution of Variables

When approaching a mathematical expression, substituting variables is often a crucial first step. This involves replacing variables in an expression with their given values. To substitute variables, follow these steps:

• Identify the variables within the expression. These are typically represented by letters such as 'a' or 'b'.
• Replace each variable with its corresponding numerical value provided in the problem.

In the exercise, we substituted 'a' with -1 and 'b' with 5. This changes the expression from \( \frac{\sqrt{86 + ab}}{a} \) to\( \frac{\sqrt{86 + (-1)(5)}}{-1} \). Substitution helps in transforming abstract expressions into ones that are easier to compute and understand.

Simplifying Expressions

Simplification involves breaking down a complex expression into a simpler form. This process can make calculations more straightforward and reduce potential errors. Here's how to simplify an expression:

• Perform any arithmetic operations like multiplication and addition under the root first.
• Combine like terms whenever possible to reduce the expression.

For the expression \( \frac{\sqrt{86 + (-5)}}{-1} \), simplifying under the square root gives \( \sqrt{81} \) by performing the arithmetic operation: \( 86 - 5 \).Simplifying makes it easier to see what operations are next, setting the stage for further calculations.

Square Roots

A square root finds a number which, when multiplied by itself, gives the original number under the root. Understanding square roots involves:

• Recognizing perfect squares (numbers like 81, which equal 9 when squared).
• Simplifying the number under the root to its square root.

In our example, the square root of 81 equates to 9 since \( 9 \times 9 = 81 \). Calculating square roots is crucial when simplifying radical expressions like the one in the problem.

Division of Negative Numbers

Dividing numbers, particularly negative ones, can change the sign of the result. Understanding this concept is key to finding the correct answer. Here are some quick tips about division:

• Dividing a positive number by a negative number results in a negative quotient.
• Similarly, dividing a negative number by a positive one yields a negative result.

In the exercise, the division of 9 by -1 results in -9, because a positive divided by a negative equals a negative. Being cautious with signs during division is essential for achieving accurate results in mathematical calculations.