/*! 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 11 Evaluate each expression. $$ ... [FREE SOLUTION] | 91影视

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Chapter 0: Problem 11

Evaluate each expression. $$ \frac{10^{7}}{10^{4}} $$

Short Answer

Expert verified
The evaluated expression is 1000.

Step by step solution

01

Applying the Quotient of Powers Property

To evaluate the expression \( \frac{10^{7}}{10^{4}} \), we can apply the quotient of powers property. This states that \( \frac{a^m}{a^n} = a^{m-n} \), provided \( a eq 0 \). Here, \( a = 10 \), \( m = 7 \), and \( n = 4 \).
02

Simplifying the Expression

Now, apply the exponents' subtraction: \( 10^{7-4} = 10^{3} \). This simplifies the expression to \( 10^{3} \).
03

Calculate the Power of Ten

The expression \( 10^{3} \) means \( 10 \) raised to the power of \( 3 \). Calculate this by multiplying three tens together: \( 10 \times 10 \times 10 = 1000 \). Thus, \( 10^{3} = 1000 \).

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Key Concepts

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

Simplifying Exponential Expressions
Simplifying exponential expressions can be much simpler if you use certain mathematical properties. One of the most useful for dealing with expressions that have the same base is the "Quotient of Powers Property". This property tells us that when we divide two powers with the same base, we can subtract the exponents:
  • If you have an expression like \( \frac{a^m}{a^n} \), you can simplify it to \( a^{m-n} \).
  • Here, the base \( a \) must not be zero, and \( m \) and \( n \) are the exponents.
In this exercise, we have \( \frac{10^7}{10^4} \), which simplifies to \( 10^{7-4} \). Hence, the expression reduces to \( 10^3 \). This illustrates how exponent properties simplify potentially complex expressions into manageable calculations.
Powers of Ten
The power of ten is an exciting and essential concept, especially when dealing with large numbers or simplifying expressions. When you see \( 10^3 \) or similar expressions, you're working with powers of ten, which follow some clear principles:
  • In an expression like \( 10^n \), the 10 is the base, and \( n \) is the exponent.
  • It represents multiplying 10 by itself \( n \) times. So, \( 10^3 \) means \( 10 \times 10 \times 10 \).
Powers of ten are especially popular because they simplify arithmetic in base-10, which is the numbering system we use daily. In this problem, the final answer \( 10^3 = 1000 \), is easy to verify by multiplying three tens in a row.
Exponent Subtraction
Exponent subtraction is crucial when simplifying fractions of exponential expressions with the same base. Using exponent subtraction, we follow these steps:
  • When dividing two exponential expressions, subtract the exponent of the denominator from the exponent of the numerator.
  • Apply it using the formula \( \frac{a^m}{a^n} = a^{m-n} \).
In this exercise, we apply this principle to the fraction \( \frac{10^7}{10^4} \), ensuring our result is \( 10^{3} \). This technique reduces complex expressions into simpler forms, making calculations straightforward.
It's a handy tool for students tackling exponential problems, as it transforms potential arithmetic challenges into user-friendly equations.

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Most popular questions from this chapter

Simplify the expression and eliminate any negative exponent(s). $$ (r s)^{3}(2 s)^{-2}(4 r)^{4} $$

The Power of Algebraic Formulas Use the Difference of Squares Formula to factor \(17^{2}-16^{2} .\) Notice that it is easy to calculate the factored form in your head, but not so easy to calculate the original form in this way. Evaluate each expression in your head: (a) \(528^{2}-527^{2} \quad\) (b) \(122^{2}-120^{2} \quad\) (c) \(1020^{2}-1010^{2}\) Now use the product formula \((A+B)(A-B)=A^{2}-B^{2}\) to evaluate these products in your head: (d) 49\(\cdot 51 \quad\) (e) 998\(\cdot 1002\)

Simplify the expression and eliminate any negative exponent(s). $$ \frac{x^{9}(2 x)^{4}}{x^{3}} $$

Simplify the expression and eliminate any negative exponent(s). $$ \left(3 y^{2}\right)\left(4 y^{5}\right) $$

Number of Molecules A sealed room in a hospital, measuring 5 \(\mathrm{m}\) wide, 10 \(\mathrm{m}\) long, and 3 \(\mathrm{m}\) high, is filled with pure oxygen. One cubic meter contains 1000 \(\mathrm{L}\) , and 22.4 \(\mathrm{L}\) of any gas contains \(6.02 \times 10^{23}\) molecules (Avogadro's number). How many molecules of oxygen are there in the room?
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