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Chapter 5: Problem 103

Simplify each expression. a) \(\left(\frac{4}{9}\right)^{-3}\) (b) \(\left(-\frac{u}{v}\right)^{-5}\)

Short Answer

Expert verified
(a) \(\frac{729}{64}\), (b) \(-\frac{v^5}{u^5}\)

Step by step solution

01

Understand Negative Exponents

A negative exponent means you take the reciprocal of the base and then apply the positive exponent. For example, \(a^{-b} = \frac{1}{a^b}\).
02

Simplify (a) \(\left(\frac{4}{9}\right)^{-3}\)

Applying the rule of negative exponents, we get: \(\left(\frac{4}{9}\right)^{-3} = \left(\frac{9}{4}\right)^3\).
03

Calculate \(\left(\frac{9}{4}\right)^3\)

Raise both the numerator and the denominator to the power of 3: \(\left(\frac{9}{4}\right)^3 = \frac{9^3}{4^3} = \frac{729}{64}\).
04

Simplify (b) \(\left(-\frac{u}{v}\right)^{-5}\)

Again, apply the rule of negative exponents: \(\left(-\frac{u}{v}\right)^{-5} = \left(-\frac{v}{u}\right)^5\).
05

Calculate \(\left(-\frac{v}{u}\right)^5\)

Raise both the negative fraction and its components to the power of 5: \(\left(-\frac{v}{u}\right)^5 = -\frac{v^5}{u^5}\).

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Key Concepts

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

Positive Exponents
Positive exponents indicate how many times a number (the base) is multiplied by itself. For example, in the expression \(a^b\), the base \(a\) is multiplied by itself \(b\) times.
  • Example: \(2^3 = 2 \times 2 \times 2 = 8\).
  • Example: \(5^2 = 5 \times 5 = 25\).
Positive exponents are straightforward and simply involve repeated multiplication. This is a key concept to understand before moving into more complex exponent rules, like negative exponents and reciprocals.
Reciprocals
Reciprocals are closely related to both positive and negative exponents. The reciprocal of a number is 1 divided by that number. If you have a fraction, its reciprocal is switching the numerator and denominator.
  • Example: The reciprocal of \(a\) is \(\frac{1}{a}\).
  • Example: The reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\).
Reciprocals are helpful when dealing with negative exponents, as they turn the base into its reciprocal. For instance, \(2^{-3}\) can be written as \(\left(\frac{1}{2}\right)^3\). It's important to become comfortable switching between a number and its reciprocal when following exponent rules.
Fractional Exponents
Fractional exponents represent roots. The numerator of the fraction is the power, and the denominator is the root. For instance, \(a^{\frac{1}{2}}\) represents the square root of \(a\), and \(a^{\frac{1}{3}}\) represents the cube root of \(a\).
  • Example: \(4^{\frac{1}{2}} = \sqrt{4} = 2\).
  • Example: \(8^{\frac{1}{3}} = \sqrt[3]{8} = 2\).
Fractional exponents let us simplify roots and exponents into a single expression, making complex calculations easier to handle. Combining these with reciprocal operations helps to grasp the transition between different exponent forms.
Negative Fractions
Negative fractions follow the same rules as positive fractions but with an added twist because of the negative sign. When a fraction has a negative exponent, we first take the reciprocal, making it easier to manage.
  • Example: \(\left(\frac{4}{9}\right)^{-3}\) simplifies as follows: Take the reciprocal to get \(\left(\frac{9}{4}\right)^3\) and then calculate to get \(\frac{729}{64}\).
  • Example: \(\left(-\frac{u}{v}\right)^{-5}\) converts first to \(\left(-\frac{v}{u}\right)^5\) and then calculates as \(-\frac{v^5}{u^5}\).
Handling negative fractions requires familiarity with reciprocals and positive exponents. Each step builds upon these foundational concepts to simplify expressions accurately.

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Most popular questions from this chapter

Multiply each pair of conjugates using the Product of Conjugates Pattern. $$ (7 w+10 x)(7 w-10 x) $$

Find each product. $$ (t-9)^{2} $$

Multiply the monomials. (a) \(\left(-8 u^{6}\right)(-9 u)\) (b) \(\left(\frac{2}{3} x^{2} y\right)\left(\frac{3}{4} x y^{2}\right)\)

Multiply each pair of conjugates using the Product of Conjugates Pattern. $$ (a b-4)(a b+4) $$

Find each product. $$ (5 x-y)(x-4) $$
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