Problem 13 - Algebra for College Students

Question

Evaluate each of the following. For example, \(\sqrt{25}=5\). \(-\sqrt{\frac{36}{49}}\)

Step-by-Step Solution

Answer

The answer is \(-\frac{6}{7}\).

Step 1: Understanding the Expression

The expression we need to evaluate is \[-\sqrt{\frac{36}{49}}\].This represents the negative square root of a fraction, \(\frac{36}{49}\).

Step 2: Evaluate the Fraction

To find the square root of the fraction \(\frac{36}{49}\), we need to find the square root of the numerator and the denominator separately.- The square root of 36 is 6, because \(6 \times 6 = 36\).- The square root of 49 is 7, because \(7 \times 7 = 49\).This gives us \(\sqrt{\frac{36}{49}} = \frac{6}{7}\).

Step 3: Apply the Negative Sign

Since there is a negative sign outside the square root, we apply it to the result from the previous step.Therefore, the final answer is \[-\sqrt{\frac{36}{49}} = -\frac{6}{7}\].

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are usually denoted by a minus sign (-) in front of them. Working with negative numbers can seem tricky at first, but understanding a few simple rules can make it easier.

• Adding a negative number is equivalent to subtracting its positive counterpart. For example, adding -5 is the same as subtracting 5.
• Multiplying two negative numbers together gives a positive result because the negatives cancel each other out.
• When multiplying a positive number by a negative number, the result is always negative.
• Similarly, dividing two negative numbers gives a positive result, while dividing a negative number by a positive number (or vice versa) results in a negative outcome.

In the context of our exercise, the negative sign in front of the square root indicates that the final result must be negative. Paying attention to such signs can help prevent errors in calculations.

Fractional Exponents

Fractional exponents are another way of expressing roots. Instead of writing the square root of a number using the radical symbol (√), we use a fractional exponent. For instance, the square root of a number can be written as the number raised to the power of 1/2. Understanding this allows us to manipulate and simplify expressions more flexibly.

• The square root of a number, such as \(\sqrt{9}\), can be rewritten using fractional exponents as \(9^{1/2}\).
• Similarly, cube roots can be expressed with a fractional exponent of 1/3, and so on for other roots.
• You can further simplify expressions with fractional exponents by converting them back to their root form.

In the exercise, knowing that the square root is equivalent to the 1/2 power could help understand how each component, such as the numerator or denominator of a fraction, is separately rooted. This flexibility is helpful when simplifying expressions or solving equations.

Simplifying Expressions

Simplifying expressions involves reducing them to a simpler form without changing their value. This makes it easier to understand and solve mathematical problems. Here's how you can break it down:

• Identify components that can be simplified separately, such as finding the square root of the numerator and the denominator separately, as seen in our exercise.
• Combine like terms and apply arithmetic operations carefully, always considering negative or positive signs attached to the numbers.
• Use properties of exponents and roots to manage and transform complex expressions.

When simplifying the given exercise, first take the square root of the fraction by evaluating the numerator and the denominator independently. This means that \( \sqrt{\frac{36}{49}} = \frac{6}{7} \ \). Finally, apply any external modifiers such as the negative sign to reach the simplest form of the expression, which, in this case, is \( -\frac{6}{7} \ \). Simplifying expressions is a valuable skill that helps in breaking down complex problems into manageable steps.