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Chapter 14: Problem 106

Assume that all exponents are integers, all denominators are nonzero, and zero is not raised to a nonpositive power. $$\left[\left(\frac{x^{r}}{y^{t}}\right)^{2}\left(\frac{x^{2 r}}{y^{4 t}}\right)^{-2}\right]^{-3}$$

Short Answer

Expert verified
The simplified expression is \(\frac{x^{6r}}{y^{18t}}\).

Step by step solution

01

Simplify the expression inside the inner brackets

We have the expression inside the inner brackets: \(\left(\frac{x^{r}}{y^{t}}\right)^{2}\left(\frac{x^{2 r}}{y^{4 t}}\right)^{-2}\) First, apply the power of a power property to each part of the expression: \(\frac{x^{2r}}{y^{2t}} \cdot \frac{x^{-4r}}{y^{-8t}}\)
02

Multiply the fractions and simplify it

Now, multiply the fractions and apply the property of multiplying powers with the same base: \(\frac{x^{2r}x^{-4r}}{y^{2t}y^{-8t}}\) Combine the exponents: \(\frac{x^{2r-4r}}{y^{2t-8t}}\) Simplify the exponents: \(\frac{x^{-2r}}{y^{-6t}}\)
03

Consider the reciprocal of a power

Since we are dealing with negative exponents, apply the reciprocal of a power to rewrite it as positive exponents: \(\frac{y^{6t}}{x^{2r}}\)
04

Apply the outer exponent to the simplified expression

Now, let's apply the outer exponent -3 to the simplified expression inside the outer brackets: \(\left[\frac{y^{6t}}{x^{2r}}\right]^{-3}\) Apply the power of a power property: \(\frac{y^{-18t}}{x^{-6r}}\)
05

Consider the reciprocal of a power again

Lastly, apply the reciprocal of a power to rewrite the negative exponents as positive exponents and simplify the expression: \(\frac{x^{6r}}{y^{18t}}\) After going through all the steps, the simplified expression is: \(\frac{x^{6r}}{y^{18t}}\)

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Key Concepts

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

Integer Exponents
Exponents are a way to express repeated multiplication of a number by itself. Integer exponents refer to exponents that are whole numbers. For instance, in the expression \( x^3 \), the exponent 3 signifies that \( x \) is multiplied by itself three times: \( x \times x \times x \). Understanding integer exponents is crucial because they appear frequently in algebraic expressions and help to simplify calculations.

Here are some key properties to remember:
  • Multiplication: When multiplying numbers with the same base, add the exponents: \( a^m \times a^n = a^{m+n} \).
  • Division: When dividing numbers with the same base, subtract the exponents: \( a^m / a^n = a^{m-n} \).
  • Power of a Power: When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{mn} \).
These rules help to simplify expressions and make intricate algebraic problems more manageable.
They are the building blocks for more complex operations involving exponents.
Negative Exponents
Negative exponents might initially seem confusing, but they are straightforward once you grasp the concept. A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent. For instance, \( a^{-n} = \frac{1}{a^n} \).

So, if you encounter a negative exponent:
  • Move the base to the reciprocal (the opposite part of a fraction) and change the sign of the exponent to positive.
  • This helps in transforming difficult expressions into more manageable forms.
For example, consider \( x^{-3} \), this can be rewritten as \( \frac{1}{x^3} \).
This process not only reduces complexity but is essential for simplification in algebra.
Power of a Power
In algebra, when you have an exponent raised to another exponent, you are dealing with the power of a power. This situation requires multiplying the exponents to find the new power.
Consider an example: \( (x^m)^n \) becomes \( x^{m \times n} \). This property allows for further simplification within algebraic expressions and is often used alongside other properties like multiplying or dividing exponents.

Application examples include:
  • Simplifying nested exponents, such as \( ((x^2)^3) = x^{2 \times 3} = x^6 \).
  • Rewriting complex algebraic terms into more manageable forms.
This principle is significant for working through problems that involve multiple layers of exponents.
Simplifying Fractions
Simplifying fractions is crucial in algebra to reduce complexity and find a clearer representation of expressions. When working with fractions involving exponents, the same basic principles apply, but with the inclusion of exponent rules.

Steps include:
  • Multiply or divide the bases: Ensuring to apply exponent rules for combination \( \frac{a^m}{a^n} = a^{m-n} \).
  • Simplify negative exponents: Use the reciprocal rules to express them as positive.
  • Reduce the fraction: Bring the terms to their simplest form for ease of understanding.
Simplifying is about making expressions easier to work with by reducing complexity.
This not only aids in solving equations but is fundamental for clarity in mathematical communication.

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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.

Solve the logarithmic equation algebraically. Then check using a graphing calculator. $$\ln x=-2$$

Natural Fibers Clothing charges \(\$ 4\) for shipping orders of \(\$ 25\) or less, \(\$ 8\) for orders from \(\$ 25.01\) to \(\$ 75,\) and \(\$ 10\) for orders over \(\$ 75 .\) One week, shipping charges for 600 orders totaled \(\$ 4280 .\) Eighty more orders for \(\$ 25\) or less were shipped than orders for more than \(\$ 75 .\) Find the number of orders shipped at each rate.

Approximate the point \((s)\) of intersection of the pair of equations. $$y=\ln 3 x, y=3 x-8$$

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 When the numerator and the denominator of a rational function have the same degree, the graph of the function has_____
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