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Chapter 0: Problem 49

Evaluate each exponential expression. $$\frac{8 x^{20}}{2 x^{4}}$$

Short Answer

Expert verified
The simplified form of \(\frac{8 x^{20}}{2 x^{4}} \) is \(4x^{16}\).

Step by step solution

01

Simplify the Coefficients of the Terms

First, divide the coefficients in front of the variables which are 8 and 2. \[ \frac{8}{2} = 4 \]
02

Subtract the Exponents of Like Bases

Since, when we divide terms with the same base, we subtract the exponents, we subtract the 4 from 20. i.e. \(20 - 4 = 16\).
03

Write the Final Answer

Combine the results from the first two steps to give the final answer as \(4x^{16}\).

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Key Concepts

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

Coefficients: The Big Numbers
Coefficients in mathematical expressions are the numerical values that stand before variables or terms. They indicate how many times you should multiply that particular term. For example, in the expression \(8x^{20}\), the coefficient is 8.
Think of coefficients as the quantity or the main multiplier of the expression they attach to:
  • If you have \(3x\), you have three of \(x\).
  • If you have \(8x^{20}\), you have eight of \(x^{20}\).
In the given problem, you retain the same base and divide the coefficients. So, dividing 8 by 2 gives 4:
  • Divide coefficients separately from variables.
  • Perform arithmetic operations just like ordinary numbers.
Thus, your simplified coefficient multiplier is 4.
Exponents: The Power to Multiply
Exponents represent how many times the base is multiplied by itself. They play a pivotal role in simplifying or evaluating expressions similar to our example. In \(x^{20}\), the exponent is 20, which tells us \(x\) is multiplied by itself 20 times.
When dealing with division of exponential terms, you use the rule of subtracting exponents when the bases are the same. This is a crucial point to grasp:
  • If you divide \(x^{a}\) by \(x^{b}\), you subtract: \(a - b\).
  • Hence, \(x^{20} \div x^{4} = x^{20 - 4} = x^{16}\).
The rule simplifies the expression significantly. Never confuse it, though, with multiplication, where you would add the exponents instead.
Like Bases: Staying Consistent
The concept of like bases is critical when simplifying expressions that involve terms with exponents. A 'base' in a term such as \(x^{n}\) is the number \(x\) raised to a power by the exponent \(n\).
When the question involves division or multiplication of exponential terms, like bases must be compared:
  • In \(\frac{8x^{20}}{2x^{4}}\), the base \(x\) is consistent in both the numerator and the denominator.
  • This consistency allows the direct application of exponent rules, such as subtracting exponents in division.
The goal is to work with like bases so that simplifying the expression is straightforward. Mastering the basics of terms with identical bases prepares you for more complex algebraic manipulations.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Although \(20 x^{3}\) appears in both \(20 x^{3}+8 x^{2}\) and \(20 x^{3}+10 x\) I'll need to factor \(20 x^{3}\) in different ways to obtain each polynomial's factorization.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. First factoring out the greatest common factor makes it easier for me to determine how to factor the remaining factor, assuming that it is not prime.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'll win the contest if I can complete the crossword puzzle in 20 minutes plus or minus 5 minutes, so my winning time, \(x,\) is modeled by \(|x-20| \leq 5\)

Will help you prepare for the material covered in the next section. Multiply and simplify: \(12\left(\frac{x+2}{4}-\frac{x-1}{3}\right)\)

A company wants to increase the \(10 \%\) peroxide content of its product by adding pure peroxide (100\% peroxide). If \(x\) liters of pure peroxide are added to 500 liters of its \(10 \%\) solution, the concentration, \(C,\) of the new mixture is given by $$C=\frac{x+0.1(500)}{x+500}$$ How many liters of pure peroxide should be added to produce a new product that is \(28 \%\) peroxide?
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