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Chapter 4: Problem 18

Simplify. $$\left(\frac{1}{10}\right)^{4}$$

Short Answer

Expert verified
The simplified form is \(\frac{1}{10000}\).

Step by step solution

01

Understand the Problem

We need to simplify the expression \(\bigg(\frac{1}{10}\bigg)^4\). This requires us to evaluate the expression without using a calculator.
02

Apply Exponent Rules

For any fraction \(\bigg(\frac{a}{b}\bigg)^n\), we raise both numerator and denominator to the power of \(n\). In this case, we raise \(1\) and \(10\) to the power of \(4\): \(\bigg(\frac{1}{10}\bigg)^4= \frac{1^4}{10^4}\).
03

Compute the Powers

Calculate the power of each term: \(1^4 = 1\) and \(10^4 = 10000\). This gives us \(\frac{1}{10000}\).
04

Write the Simplified Form

So, the simplified form of \(\bigg(\frac{1}{10}\bigg)^4\) is \(\frac{1}{10000}\).

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Key Concepts

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

Simplifying Fractions
Fractions can look complicated, but the process of simplifying them can make them much easier to work with. Here's how you can approach it:
  • First, understand the numerator (the top part) and the denominator (the bottom part).
  • If both the numerator and the denominator can be divided by the same number, you can reduce or simplify the fraction.
For example, \(\frac{4}{8}\) can be simplified by dividing both the numerator and the denominator by 4, resulting in \(\frac{1}{2}\).
However, in the given exercise, \(\left(\frac{1}{10}\right)^{4}\), we are dealing with an expression that requires more complex steps involving exponents.
Exponent Rules
Exponentiation is one of the fundamental mathematical operations, and there are specific rules that govern how exponents work. When raising a fraction to a power, make sure you apply the exponent to both the numerator and the denominator.
Some key exponent rules include:
  • Product of powers: \(a^m \cdot a^n = a^{m+n}\)
  • Power of a power: \(\left(a^m\right)^n = a^{m \cdot n}\)
  • Power of a fraction: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)
In the given exercise, \(\left(\frac{1}{10}\right)^{4}\), we use the power of a fraction rule, applying the exponent 4 to both 1 and 10. This gives us \(\frac{1^4}{10^4}\).
Powers of Numbers
Understanding powers of numbers is crucial for solving a variety of mathematical problems. When you raise a number to a power, you are multiplying that number by itself for the number of times indicated by the exponent.
For example:
  • \(2^3 = 2 \cdot 2 \cdot 2 = 8\)
  • \(5^2 = 5 \cdot 5 = 25\)
In our specific case from the exercise, we need to calculate \(1^4\) and \(10^4\):
  • \(1^4\) is straightforward because any number to the power of 4 where the base number is 1 remains 1 (\(1 \cdot 1 \cdot 1 \cdot 1 = 1\)).
  • \(10^4\) means multiplying 10 by itself four times, giving us \(10 \cdot 10 \cdot 10 \cdot 10 = 10{,}000\).
Putting these results together, \(\left(\frac{1}{10}\right)^{4}\) simplifies to \(\frac{1}{10{,}000}\).

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Most popular questions from this chapter

Simplify to lowest terms by first reducing the powers of 10. $$\frac{42,000}{22,000}$$

Simplify. $$\left|\frac{1}{10}\right|$$

Simplify to lowest terms by first reducing the powers of 10. $$\frac{5100}{30,000}$$

Convert the improper fraction to a mixed number. $$-\frac{27}{10}$$

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