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Chapter 0: Problem 15

\(7-28\) Evaluate each expression. $$ \left(\frac{1}{3}\right)^{4}(-3)^{2} $$

Short Answer

Expert verified
The evaluated expression is \( \frac{1}{9} \).

Step by step solution

01

Evaluate the Power of the Rational Number

Calculate the value of \( \left(\frac{1}{3}\right)^4 \). This means you need to multiply \( \frac{1}{3} \) by itself 4 times: \( \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{81} \).
02

Evaluate the Power of the Negative Integer

Calculate the value of \( (-3)^2 \). Since the exponent is 2, this is equivalent to \( -3 \times -3 = 9 \).
03

Multiply the Results

Multiply the results obtained from the previous steps: \( \frac{1}{81} \times 9 \). This gives \( \frac{9}{81} \).
04

Simplify the Fraction

Simplify the fraction \( \frac{9}{81} \). Both the numerator and the denominator can be divided by 9, so \( \frac{9}{81} = \frac{1}{9} \).

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Key Concepts

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

Powers of Fractions
When we talk about powers of fractions, we're examining how a fraction behaves when it is raised to an exponent. Think of it this way: when you see something like \( \left( \frac{1}{3} \right)^4 \), you need to multiply \( \frac{1}{3} \) by itself multiple times. Specifically, you multiply it by itself 4 times:
  • Start with \( \frac{1}{3} \).
  • Multiply it: \( \frac{1}{3} \times \frac{1}{3} = \frac{1}{9} \).
  • Multiply again: \( \frac{1}{9} \times \frac{1}{3} = \frac{1}{27} \).
  • Once more: \( \frac{1}{27} \times \frac{1}{3} = \frac{1}{81} \).
Each time you multiply, you are essentially making the fraction smaller, because the base is less than 1. Powers of fractions shrink, rather than grow, as more multiples are involved since the values remain under 1. Remember, understanding this concept will make complicated-looking problems much simpler to tackle.
Powers of Negative Numbers
Negative numbers follow specific rules when it comes to exponents. This is important because sometimes error can occur if we forget about these rules. When a negative number is raised to an even power, like in \((-3)^2\), the result is positive. Why? Because:
  • \( -3 \times -3 = 9 \).
  • A negative times a negative is always positive.
However, if a negative number is raised to an odd power, the result remains negative. Imagine \((-3)^3\):
  • \( -3 \times -3 = 9 \)
  • Then, \(9 \times -3 = -27\).
Always remember:
  • Even exponents of negative numbers yield positive results.
  • Odd exponents keep the result negative.
These patterns are key to mastering powers of negative numbers.
Simplifying Fractions
Simplifying fractions is like cleaning them up to make them look neater and easier to read. After performing operations with fractions, like multiplication, the result might be a fraction that can be reduced further. For instance, \( \frac{9}{81} \) simplifies to \( \frac{1}{9} \).Here鈥檚 the process:
  • Look for the greatest common factor (GCF) of both the numerator and the denominator.
  • Divide both by this common factor. In this case, 9 is the GCF of both 9 and 81.
  • Divide: \( \frac{9}{9} = 1 \) and \( \frac{81}{9} = 9 \), leading to \( \frac{1}{9} \).
This operation makes it easier to work with and understand the fraction. Simplifying is crucial because it reduces fractions to their simplest form, making calculations more straightforward and less prone to errors.

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Most popular questions from this chapter

Simplify the compound fractional expression. $$ \frac{x+\frac{1}{x+2}}{x-\frac{1}{x+2}} $$

Simplify the compound fractional expression. $$ \frac{\frac{x}{y}-\frac{y}{x}}{\frac{1}{x^{2}}-\frac{1}{y^{2}}} $$

State whether the given equation is true for all values of the variables. (Disregard any value that makes a denominator zero.) $$ \frac{16+a}{16}=1+\frac{a}{16} $$

Perform the addition or subtraction and simplify. $$ \frac{1}{x^{2}+3 x+2}-\frac{1}{x^{2}-2 x-3} $$

Simplify the rational expression. $$ \frac{14 t^{2}-t}{7 t} $$
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