/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 69 Evaluate the expression. $$ ... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 69

Evaluate the expression. $$ 10^{-4} $$

Short Answer

Expert verified
\(\frac{1}{10,000}\)

Step by step solution

01

Understanding the concept of negative exponents

A negative exponent rule states that any non-zero number raised to a negative power equals its reciprocal raised to the absolute value of the exponent. In mathematical terms, \(a^{-n} = \frac{1}{a^n}\) where 'a' is any non-zero number and 'n' is a positive integer.
02

Apply the negative exponent rule

According to the rule, the expression \(10^{-4}\) can be rewritten as \(\frac{1}{10^4}\)
03

Calculate the positive power

The expression \(10^4\) equals 10,000. In mathematical terms, \(10^4 = 10 \times 10 \times 10 \times 10 = 10,000\)
04

Complete the calculation

Now, replace \(10^4\) with 10,000 in the expression. So the result equals \(\frac{1}{10,000}\)

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Key Concepts

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

Reciprocal
Understanding the concept of reciprocals is key when dealing with negative exponents. In mathematics, the reciprocal of a number is simply one divided by that number. For instance, the reciprocal of 5 is \(\frac{1}{5}\). This applies to negative exponents because they flip or invert the base number. For an expression like \(10^{-4}\), its reciprocal involves converting the base, 10, into \(\frac{1}{10^4}\). It's like a mirror image that switches a negative exponent into a positive one in the denominator. Reciprocals are very useful in simplifying expressions and solving equations in mathematics.
Exponential Expressions
Exponential expressions involve numbers raised to a power. The number being multiplied is called the base, and the power represents how many times the base is used as a factor. For example, in \(3^2\), 3 is the base, and 2 is the exponent, indicating that 3 is multiplied by itself to get 9. When you encounter negative exponents, they tell us to take the reciprocal of the base and then apply the positive form of the exponent. So, \(a^{-n}\) would be converted to \(\frac{1}{a^n}\).Exponential expressions are foundational to many areas in mathematics, making it crucial to understand both positive and negative exponents.
Powers of Ten
Powers of ten are especially easy to handle due to the decimal system. When you raise 10 to a positive power like \(10^4\), it means you multiply 10 by itself 4 times, resulting in 10,000. The more powerful aspect of powers of ten is the simplicity it brings when written, as each power corresponds to a specific number of zeros.Negative powers of ten mean you're dividing instead of multiplying. For example, \(10^{-4}\) turns into \(\frac{1}{10,000}\), which is the same as moving the decimal point four places to the left. Utilizing the powers of ten concept simplifies calculations and is essential for scientific notation used in expressing very large or small numbers.

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