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Chapter 14: Problem 34

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

Short Answer

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

Step by step solution

01

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.
02

Rewrite the expression using the rule for negative exponents

We can rewrite the given expression using the rule for negative exponents: \[ \frac{17^{-18}}{11^{-13}} = \frac{1}{17^{18}} \cdot \frac{11^{13}}{1}. \]
03

Multiply fractions

Now we can multiply the fractions: \[ \frac{1}{17^{18}} \cdot \frac{11^{13}}{1} = \frac{11^{13}}{17^{18}}. \]
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

In New York City, orange juice, a raisin bagel, and a cup of coffee from Katie's Koffee Kart cost a total of \(\$ 8.15 .\) Katie posts a notice announcing that, effective the following week, the price of orange juice will increase \(25 \%\) and the price of bagels will increase \(20 \% .\) After the increase, the same purchase will cost a total of \(\$ 9.30\), and the raisin bagel will cost \(30 鈧琝) more than coffee. Find the price of each item before the increase.

Fill in the blank with the correct term. Some of the given choices will not be used. Descartes' rule of signs a vertical asymptote the leading-term test \(\quad\) an oblique the intermediate \(\quad\) asymptote value theorem direct variation the fundamental \(\quad\) inverse variation theorem of algebra a horizontal line a polynomial function a vertical line a rational function \(\quad\) parallel a one-to-one function \(\quad\) perpendicular a constant function a horizontal asymptote _____is a function that is a quotient of two polynomials.

The top three Mother's Day gifts are flowers, jewelry, and gift certificates. The total of the average amounts spent on these gifts is \(\$ 53.42 .\) The average amount spent on jewelry is \(\$ 4.40\) more than the average amount spent on gift certificates. Together, the average amounts spent on flowers and gift certificates is \(\$ 15.58\) more than the average amount spent on jewelry. (Source: BIG research) What is the average amount spent on each type of gift?

Solve the system of equations. $$\begin{aligned} w+x-y+z &=0 \\ -w+2 x+2 y+z &=5 \\ -w+3 x+y-z &=-4 \\ -2 w+x+y-3 z &=-7 \end{aligned}$$

Solve the logarithmic equation algebraically. Then check using a graphing calculator. $$\log _{5} x=4$$
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