Problem 66 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Simplify the expression. $$ \frac{6^8}{6^{10}} $$

Step-by-Step Solution

Answer

\(\frac{1}{36}\)

Step 1: Identify the like bases

We have \(6^8\) and \(6^{10}\) which indeed share a common base, which is number 6.

Step 2: Apply the exponent subtraction rule

When you're dividing like bases, subtract the exponent in the denominator from the exponent in the numerator. So it will look like this: \(6^{8-10}\)

Step 3: Simplify

Perform the subtraction operation: \(6^{-2}\)

Step 4: Rewrite using a reciprocal

A negative exponent indicates a reciprocal. A reciprocal of x is 1/x. So \(6^{-2}\) can be simply written as \(\frac{1}{6^2}\).

Step 5: Final Simplification

So \(\frac{1}{6^2}\) is equal to \(\frac{1}{36}\).

Key Concepts

Exponent Subtraction Rule

When dealing with exponential expressions, it's essential to understand the exponent subtraction rule for simplification. This rule comes into play when you have the same base raised to different powers and are dividing them. For example, in the expression \(\frac{6^8}{6^{10}}\), both 6^8 and 6^10 have the same base of 6.

To apply the rule effectively, simply subtract the exponent in the denominator from the exponent in the numerator. In our case, it looks like this: \(6^{8-10}\). This operation helps in reducing the expression to a form which is easier to manage and interpret.

Understanding this rule is not only crucial for the simplification process but also enhances your algebraic manipulation skills. It's widely applicable and should be used whenever like bases appear in a division format.

Negative Exponents

The concept of negative exponents might seem intimidating at first, but it represents a straightforward principle. A negative exponent indicates the reciprocal of the base raised to the absolute value of the given exponent. For example, \(6^{-2}\) signifies the reciprocal of 6 raised to the second power.

By definition, \(a^{-n} = \frac{1}{a^n}\), where a is any nonzero number and n is a positive integer. This also means that multiplying a number by its negative exponent is equivalent to one over the number raised to that respective positive exponent.

In practice, this concept is a powerful tool for simplifying expressions and solving equations that involve exponential terms. It also helps in understanding the foundational aspects of exponential growth and decay in more complex applications like science and finance.

Reciprocal of a Number

Grasping the idea of the reciprocal of a number is key to working with negative exponents and division involving exponents. The reciprocal of a number x is simply \(\frac{1}{x}\). When you have a negative exponent, you're actually dealing with the reciprocal of the base raised to a positive exponent.

For instance, in our original problem with \(6^{-2}\), the reciprocal rule informs us that this expression is equivalent to \(\frac{1}{6^2}\). The transition from a negative exponent to its reciprocal form is necessary for the final stages of simplification of expressions.

Understanding reciprocals is essential when delving into concepts such as rational expressions, complex fractions, and solving equations that entail inversion of terms. It's a fundamental aspect that reinforces mathematical fluency across various topics.