Problem 9 - Intermediate Algebra

Question

For Problems \(1-30\), evaluate each numerical expression. $$ 36^{-\frac{1}{2}} $$

Step-by-Step Solution

Answer

The value of the expression is \(\frac{1}{6}\).

Step 1: Understanding Negative Powers

To solve \(36^{-\frac{1}{2}}\), we need to understand what a negative power means. A negative exponent indicates that we take the reciprocal of the base raised to the positive of that power. Therefore, \(a^{-n} = \frac{1}{a^n}\). So, we rewrite \(36^{-\frac{1}{2}}\) as \(\frac{1}{36^{\frac{1}{2}}}\).

Step 2: Evaluating the Square Root

The next step is to evaluate \(36^{\frac{1}{2}}\). The power of \(\frac{1}{2}\) represents the square root. Therefore, \(36^{\frac{1}{2}}\) is equivalent to \(\sqrt{36}\). Since \(\sqrt{36} = 6\), we have \(36^{\frac{1}{2}} = 6\).

Step 3: Finding the Reciprocal

Now that we know \(36^{\frac{1}{2}} = 6\), we use the reciprocal that we set up in the first step to find \(36^{-\frac{1}{2}}\). Therefore, \(36^{-\frac{1}{2}} = \frac{1}{6}\).

Key Concepts

Reciprocal

The concept of a reciprocal is fundamental when dealing with negative exponents. But what is a reciprocal exactly? In simple terms, the reciprocal of a number is 1 divided by that number. For example, the reciprocal of 5 is \( \frac{1}{5} \). This concept becomes important when handling negative exponents, like in the problem of calculating \( 36^{-\frac{1}{2}} \). When we encounter a negative exponent, we flip the base to the denominator, which essentially means finding its reciprocal.When you see something like \( a^{-n} \), it transforms into \( \frac{1}{a^n} \). This means you're taking the reciprocal of \( a \) and then raising it to a positive power instead. This transformation helps manage the negative sign more effectively. Once you've handled the reciprocal, you can proceed to evaluate other components of the expression.

Square Roots

Square roots are a specific type of root that answers the question: what number times itself will equal the original number? In mathematical terms, the square root of a number \( x \) is a number \( y \) such that \( y^2 = x \). For instance, the square root of 36 is 6 because \( 6 \times 6 = 36 \).Numerically, square roots are often represented in a fractional exponent form. The expression \( x^{\frac{1}{2}} \) is another way to denote the square root of \( x \). This is why in our problem, \( 36^{\frac{1}{2}} \) means \( \sqrt{36} \). Recognizing these different notations is crucial for simplifying expressions that involve square roots, especially in the context of dealing with exponents.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form, making them easier to understand and work with. In the context of exponents and roots, simplification might involve several steps with arithmetic operations and mathematical rules.Our numerical expression, \( 36^{-\frac{1}{2}} \), simplifies through understanding negative exponents and square roots. First, handle the negative exponent by taking the reciprocal, transforming the expression to \( \frac{1}{36^{\frac{1}{2}}} \). Next, transform the fractional power to its root form, giving us \( \frac{1}{\sqrt{36}} \).

• Acknowledge the equivalent value of the square root, turning \( \sqrt{36} \) into 6.
• Substitute this back, leading to \( \frac{1}{6} \).

The key is step-by-step breaking down of complex parts, following the rules of exponents and roots, to arrive at a simplified answer.