Problem 67 - Algebra 1: Concepts and Skills

Question

Evaluate the radical expression when $a=-1 \text { and } b=5. $$\frac{\sqrt{b^{2}-75 a}}{b}$$

Step-by-Step Solution

Answer

The evaluated expression, when \(a=-1\) and \(b=5\) is 2.

Step 1: Substitute the values of 'a' and 'b'

Substitute 'a' with -1 and 'b' with 5 in the expression. The expression becomes \(\frac{\sqrt{5^{2}-75 (-1)}}{5}\)

Step 2: Simplify inside the square root

Simplify by solving the expression under the square root. \(5^{2}\) will be 25 and \(-75 \times -1\) will be 75. This leads to \(\frac{\sqrt{25+75}}{5}\)

Step 3: Further Simplify

Add the numbers inside the square root to get 100: \(\frac{\sqrt{100}}{5}\)

Step 4: Find the square root and simplify

The square root of 100 is 10, hence, the expression simplifies to \(\frac{10}{5}\)

Step 5: Simplify

Simplify the fraction \(\frac{10}{5}\) to get the final result of 2.

Key Concepts

Substitute Variables in Expressions

Understanding how to substitute variables in expressions is a fundamental skill in algebra. This process involves replacing variables, or unknowns, with given numbers or other expressions. For instance, if we have an expression that includes variables like 'a' and 'b', we can replace them with numerical values to find the actual value of the expression.

To accurately substitute variables:

• Identify which variable corresponds to which number or expression you want to substitute.
• Replace the variable with its given value carefully, ensuring you maintain any negative signs or operations that are attached to the variable.
• Rewrite the expression with the numerical values in place of the variables.

For example, when given the expression \(\sqrt{b^{2}-75a}\) and told that 'a' is -1 and 'b' is 5, we substitute -1 for 'a' and 5 for 'b'. This changes the expression into a numeric one, which can then be simplified.

Simplifying Square Roots

Simplifying square roots involves finding the principal square root of a number, which is the non-negative root that can be derived from a perfect square. This can often make expressions shorter and easier to work with.

Here's how you simplify a square root:

• Find the largest perfect square that divides evenly into the number under the square root sign.
• Break down the square root into a multiplication of two roots - one being the square root of the perfect square you found.
• Take the square root of the perfect square, which will be a whole number, and move it outside the square root sign.
• Simplify any remaining expression outside and under the square root sign.

When we apply this to our given problem, we see \(\sqrt{100}\), where 100 is a perfect square (\(10^2\)). Therefore, the square root of 100 is simply 10, which noticeably simplifies the initial expression.

Solving Algebraic Expressions

Solving algebraic expressions involves performing a series of operations to simplify the expressions and find their value. This process can include combining like terms, applying the distributive property, and carrying out the order of operations (PEMDAS/BODMAS).

The steps to solve an algebraic expression are typically as follows:

• Substitute any given values for the variables present.
• Simplify terms inside parentheses or under radical signs.
• Perform multiplication or division as applicable.
• Add or subtract terms to find the simplified form of the expression.

In our exercise, we are presented with \(\frac{\sqrt{b^{2}-75a}}{b}\) and asked to solve for specified values of 'a' and 'b'. We follow a structured approach to simplify within the radical, find the square root, and then divide as the last step to achieve the final simplified value.