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Chapter 1: Problem 17

Evaluate each exponential expression. $$\left(-\frac{1}{3}\right)^{2}$$

Short Answer

Expert verified
\(\frac{1}{9}\)

Step by step solution

01

Understand the Base and Exponent

Identify the base and the exponent in the given expression \( \left(-\frac{1}{3}\right)^{2} \). The base is \(-\frac{1}{3}\) and the exponent is 2.
02

Apply the Exponent

Raise the base \(-\frac{1}{3}\) to the power of 2. This means you will multiply \(-\frac{1}{3}\) by itself: \( \left(-\frac{1}{3}\right) \times \left(-\frac{1}{3}\right) \).
03

Multiply the Fractions

Perform the multiplication: \( - \frac{1}{3} \times - \frac{1}{3} = \frac{1}{9} \). When you multiply two negative numbers, the result is positive.

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Key Concepts

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

exponents
Exponents tell us how many times to multiply a number (the base) by itself. In the expression \( \left( -\frac{1}{3} \right)^2 \), the base is \( -\frac{1}{3} \) and the exponent is 2. When dealing with exponents, it's important to understand that the exponent applies to the entire base.

This means \( \left( -\frac{1}{3} \right)^2 \) isn't the same as \( -\frac{1}{3^2} \). Instead, it means multiplying \( -\frac{1}{3} \) by itself, two times.
negative fractions
Negative fractions can look tricky, but they're just like regular fractions with a negative sign in front. In our exercise, the fraction base is \( -\frac{1}{3} \). When you're dealing with negative fractions, it's important to keep track of the negative sign.

Notice that multiplying two negative numbers gives a positive result. So, in the expression \( \left( -\frac{1}{3} \right)^2 \), when we multiply \( -\frac{1}{3} \) by \( -\frac{1}{3} \), the negatives cancel out, giving us a positive fraction.
multiplication of fractions
Multiplying fractions is straightforward if you remember a couple of key points. Use the formula: \( \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \).

Take the numerators (top numbers) and multiply them together. Then, multiply the denominators (bottom numbers) together. In our example:
  • Numerators: \(-1 \times -1 = 1\)
  • Denominators: \(3 \times 3 = 9\)

Combine them: \( \frac{1}{9} \). And that's your result!

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Most popular questions from this chapter

Simplify each expression. $$ n^{2}-6 m-\left(-n^{2}-2 m\right) $$

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