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Chapter 6: Problem 95

Simplify each expression with exponents. (a) \(-\left(\frac{2}{3}\right)^{2}\) (b) \(\left(-\frac{2}{3}\right)^{2}\)

Short Answer

Expert verified
a) \(-\frac{4}{9}\), b) \left(\frac{4}{9}\right)\.

Step by step solution

01

Simplify part (a)

Evaluate the expression \(-\left(\frac{2}{3}\right)^{2}\). First, calculate \(\left(\frac{2}{3}\right)^{2}\). This means \(\left(\frac{2}{3}\right) \times \left(\frac{2}{3}\right) = \frac{2 \times 2}{3 \times 3} = \frac{4}{9}\). Then apply the negative sign: \(-\frac{4}{9}\).
02

Simplify part (b)

Evaluate the expression \(\left(-\frac{2}{3}\right)^{2}\). This means \(\left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right)\). Multiplying two negative fractions results in a positive product: \(\frac{(-2) \times (-2)}{3 \times 3} = \frac{4}{9}\).

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Key Concepts

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

Understanding Exponents
Exponents are a way to express repeated multiplication of a number by itself. For example, when you see \(2^3\), it means \(2 \times 2 \times 2\). In general, for any number \('a'\) and positive integer \('n'\), the expression \(a^n\) can be written as \('a'\) multiplied by itself \('n'\) times.
Let's break it down for the given exercise:
  • For \(-\left(\frac{2}{3}\right)^2\), compute \(\left(\frac{2}{3}\right)^2\) first: \left( \frac{2}{3} \times \frac{2}{3} = \frac{4}{9} \right). Then apply the negative sign, resulting in \(-\frac{4}{9}\).
  • For \(\left(-\frac{2}{3}\right)^2\), the negative fraction multiplies itself: \left( -\frac{2}{3} \times -\frac{2}{3} = \frac{4}{9} \right). Since two negative signs make a positive, the answer is \frac{4}{9}\.
Working with Negative Numbers
Negative numbers can often be tricky, especially when combined with exponents. Remember these rules when handling negative numbers and exponents:
  • If the exponent is even, the result will be positive. For instance, \left(-2\right)^2 = 4\, because multiplying two negatives gives a positive.
  • If the exponent is odd, the result will be negative. For instance, \left(-2\right)^3 = -8\, because multiplying three negatives preserves the sign.
  • When a negative is outside the exponent, like \(-\left(a^n\right)\), compute the exponent first, then apply the negative sign. Example: \left(-2\right)^3 = -8\ because \left(2^3 = 8\ and then make it negative).
Understanding and Simplifying Fractions
Fractions represent a part of a whole and are written as \(\frac{numerator}{denominator}\). When simplifying exponents involving fractions, follow these steps:
  • Square both the numerator and the denominator. For \( \left( \frac{2}{3} \right)^2 \), it's essential to first compute: \( \left( \frac{2}{3} \right)^2 = \frac{2\times 2}{3 \times 3} = \frac{4}{9} \).
  • If the fraction is negative, pay attention to whether the negative sign is inside the exponent or outside. This affects the final sign of your result.
  • Always simplify the fraction if possible. In our examples, \( \frac{4}{9} \) is already simplified.
Mastering these fraction rules will help in simplifying expressions with exponents effectively.

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Most popular questions from this chapter

Divide the monomials. $$\frac{\left(-18 p^{4} q^{7}\right)\left(-6 p^{3} q^{8}\right)}{-36 p^{12} q^{10}}$$

Simplify. (a) \(5^{-3}\) (b) \(10^{-5}\)

When Drake simplified \(-3^{0}\) and \((-3)^{0}\) he got the same answer. Explain how using the Order of Operations correctly gives different answers.

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