Problem 49 - Elementary and Intermediate Algebra

Question

Simplify. \(\left(\frac{1}{6}\right)^{-2}\)

Step-by-Step Solution

Answer

The expression simplifies to 36.

Step 1: Understand Negative Exponents

The expression \(left(rac{1}{6} ight)^{-2}\) involves a negative exponent. A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent. So, \(a^{-n} = \frac{1}{a^n}\).

Step 2: Reciprocal of the Base

Since \(left(rac{1}{6} ight)^{-2}\) has a base of \(\frac{1}{6}\), we first find the reciprocal. The reciprocal of \(\frac{1}{6}\) is 6.

Step 3: Apply Positive Exponent

Now apply the positive exponent to the reciprocal. The expression becomes \(6^2\).

Step 4: Calculate Power

Calculate \(6^2\) by multiplying 6 by itself: \(6 \times 6 = 36\).

Step 5: Conclusion

So, \(left(rac{1}{6} ight)^{-2}\) simplifies to 36.

Key Concepts

Understanding Reciprocals

When you come across mathematical expressions, it's important to understand the concept of reciprocals. A reciprocal is essentially a "flipped-over" version of a number or a fraction. For example, the reciprocal of the fraction \( \frac{1}{6} \) is 6. This is because when we swap the numerator and the denominator, 1 becomes the denominator and 6 becomes the numerator. Simply put, multiplying a number by its reciprocal will always give you the product of 1. This property is particularly useful when dealing with negative exponents, a topic we'll dive into next.

When faced with negative exponents, remembering this reciprocal property can make solving expressions much simpler. By converting the base into its reciprocal and changing the-negative exponent to a positive one, the tasks become a straightforward calculation.

The Concept of Power of a Number

The power of a number refers to how many times a number is multiplied by itself. In mathematical notation, this is expressed as an exponent. For example, when we see \(6^2\), it tells us to multiply 6 by itself, which means \(6 \times 6\). Understanding powers of numbers is foundational in simplifying mathematical expressions, especially when an exponent is involved.

Negative exponents exhibit an interesting behavior. Instead of repeating multiplication, they tell us to take the reciprocal of the base number and apply the positive exponent. So, for an expression like \( \left(\frac{1}{6}\right)^{-2} \), the negative exponent signals us to first find the reciprocal of \( \frac{1}{6} \) and then raise it to the power of 2, resulting in \( 6^2 \) or 36.

• Positive exponents mean repeated multiplication.
• Negative exponents involve reciprocals and the same repeated multiplication applied after finding the reciprocal.
• This duality is a key part of mastering exponentiation.

Simplification of Expressions

Simplification is the process of transforming a complex expression into a more manageable or familiar form. By practicing simplification, you not only tidy up mathematical expressions but also make them easier to compute and understand.

Let's consider the expression \( \left(\frac{1}{6}\right)^{-2} \). To simplify this:

• First, acknowledge the negative exponent, which calls for finding the reciprocal of the base.
• Second, rewrite the expression with the reciprocal as the new base and a positive exponent: \( 6^2 \).
• Finally, perform the arithmetic by multiplying 6 with itself, resulting in the simplified result of 36.

By breaking down the expression into understandable parts, simplification turns intimidating-looking problems into elementary arithmetic.

Embrace simplification as a valuable skill, as it enhances problem-solving efficiency and deepens your understanding of mathematical concepts.