Question

if 260,000,000,000 is rewritten in the form c × 10ⁿ, where 1 ≤ c < 10 and n is an integer, what is the value of c?

1. -/1 points

use the rules of exponents to simplify the expression.
\\(\frac{x^{5n}y^{n - 8}}{x^{4n + 2}y^7}\\)

Explanation:

First Question (Value of \( c \)):

Step1: Rewrite the number in scientific notation

We need to express \( 260,000,000,000 \) in the form \( c\times10^{n} \) where \( 1\leq c < 10 \) and \( n \) is an integer.
To do this, we move the decimal point to the left until we get a number between 1 and 10.
Starting with \( 260,000,000,000 \), if we move the decimal point 11 places to the left, we get \( 2.6 \).
So, \( 260,000,000,000=2.6\times 10^{11} \).

Step2: Identify the value of \( c \)

From the scientific notation \( 2.6\times 10^{11} \), we can see that \( c = 2.6 \).

Step1: Use the quotient rule for exponents (\( \frac{a^{m}}{a^{k}}=a^{m - k} \)) for \( x \) terms

For the \( x \) terms, we have \( \frac{x^{5n}}{x^{4n + 2}} \). Using the quotient rule of exponents, we subtract the exponents:
\( x^{5n-(4n + 2)}=x^{5n-4n - 2}=x^{n-2} \)

Step2: Use the quotient rule for exponents for \( y \) terms

For the \( y \) terms, we have \( \frac{y^{n - 8}}{y^{7}} \). Using the quotient rule of exponents, we subtract the exponents:
\( y^{(n - 8)-7}=y^{n-8 - 7}=y^{n - 15} \)

Step3: Combine the results

Combining the simplified \( x \) and \( y \) terms, we get:
\( x^{n-2}y^{n - 15} \)

Answer:

\( c = 2.6 \)

Second Question (Simplify the expression \(\frac{x^{5n}y^{n - 8}}{x^{4n+2}y^{7}}\)):