Problem 34 - Beginning and Intermediate Algebra

Question

Simplify. $$\frac{17^{-18}}{11^{-13}}$$

Step-by-Step Solution

Answer

The simplified expression is: \(\frac{17^{-18}}{11^{-13}} = \frac{11^{13}}{17^{18}}\).

Step 1: Recall the rules of exponents

Recall that the expression \(a^{-n}\) can be rewritten as \(\frac{1}{a^n}\), and when simplifying fractions that involve exponents with the same base, you can use the formula \(\frac{a^m}{a^n} = a^{m-n}\). We will use these rules to simplify the expression.

Step 2: Rewrite the expression using the rule for negative exponents

We can rewrite the given expression using the rule for negative exponents: \[ \frac{17^{-18}}{11^{-13}} = \frac{1}{17^{18}} \cdot \frac{11^{13}}{1}. \]

Step 3: Multiply fractions

Now we can multiply the fractions: \[ \frac{1}{17^{18}} \cdot \frac{11^{13}}{1} = \frac{11^{13}}{17^{18}}. \]

Step 4: Write the final answer

The simplified expression is: \[ \frac{17^{-18}}{11^{-13}} = \frac{11^{13}}{17^{18}}. \] The exercise is simplified, and there are no further steps we can take to simplify it further.

Key Concepts

Rules of Exponents

Exponents are powerful tools in mathematics, and understanding their rules is key to simplifying expressions like \(\frac{17^{-18}}{11^{-13}}\). One critical rule to remember is that a negative exponent means the reciprocal of the base raised to the corresponding positive exponent. For any non-zero number \(a\) and any integer \(n\), the rule \(a^{-n} = \frac{1}{a^n}\) applies.

Another important rule is the quotient of powers property, which states \(\frac{a^m}{a^n} = a^{m-n}\). This rule is used when both the numerator and denominator have the same base.

• If you see \(2^{-3}\), rewrite it as \(\frac{1}{2^3}\).
• For \(\frac{a^5}{a^3}\), simplify to \(a^{5-3} = a^2\).

These rules allow for the simplification of expressions efficiently, avoiding stress and reducing complexity.

Negative Exponents

Negative exponents might look unusual at first, but they follow consistent patterns. A negative exponent, as seen in expressions like \(17^{-18}\), suggests that we should take the reciprocal of the base. Therefore, \(17^{-18}\) transforms to \(\frac{1}{17^{18}}\).

When dealing with negative exponents:

• Remember that the negative sign is an indicator to "flip" the base into its reciprocal.
• Convert \(11^{-13}\) similarly to \(\frac{1}{11^{13}}\), as seen in the problem.
• This conversion simplifies exponents and aids in further mathematical manipulations.

By flipping bases using negative exponents, expressions become more manageable and compound operations can be executed with relative simplicity.

Simplifying Fractions with Exponents

Simplifying fractions with exponents involves applying rules appropriately to combine or reduce expressions to their simplest forms. Consider the expression given in the original problem: \(\frac{17^{-18}}{11^{-13}}\).

To simplify, utilize the rule for negative exponents first. Rewrite as \(\frac{1}{17^{18}}\) and multiply by \(11^{13}\), obtained from flipping the denominator. This results in:

• A multiplied fraction \(\frac{11^{13}}{17^{18}}\).
• No further simplification as bases differ.

Simplification is complete when exponents are positive and no common base remains between numerator and denominator.