$\frac{a^{2} a^{-5}}{a^{-3} a^{0}} \cdot\left(\frac{a^{2}}{a^{3}}\right)^{-4}=a^{x}$ $\newline$ What is the value of $x$ ?

View step-by-step help

Home

Math Problems

Algebra 1

Compare linear and exponential growth

Full solution

\frac{a^{2} a^{-5}}{a^{-3} a^{0}} \cdot\left(\frac{a^{2}}{a^{3}}\right)^{-4}=a^{x}

\newline

What is the value of

x

Simplify Numerator: We will simplify the expression step by step using the properties of exponents. $\newline$ First, we simplify the numerator and the denominator separately. $\newline$ $a^{2}a^{-5}$ simplifies to $a^{2-5}$ by using the property $a^{m}a^{n} = a^{m+n}$ .
Simplify Denominator: Now we simplify $a^{2-5}$ which equals $a^{-3}$ .
Divide Numerator by Denominator: Next, we simplify the denominator $a^{-3}a^{0}$ . Since any number to the power of $0$ is $1$ , $a^{0} = 1$ . So, we have $a^{-3}\cdot 1$ which is just $a^{-3}$ .
Simplify Second Part: Now we divide the simplified numerator by the simplified denominator. $a^{-3} / a^{-3}$ equals $a^{-3-(-3)}$ by using the property $a^{m} / a^{n} = a^{m-n}$ .
Raise to Power: Simplifying $a^{(-3-(-3))}$ gives us $a^{0}$ , because $-3 - (-3)$ equals $0$ . $\newline$ Since any number to the power of $0$ is $1$ , $a^{0} = 1$ .
Multiply Results: Now we need to simplify the second part of the expression $((a^{2})/(a^{3}))^{-4}$ . First, we simplify inside the parentheses: $a^{2} / a^{3}$ equals $a^{2-3}$ by using the property $a^{m} / a^{n} = a^{m-n}$ .
Final Simplification: Simplifying $a^{2-3}$ gives us $a^{-1}$ .
Final Simplification: Simplifying $a^{(2-3)}$ gives us $a^{-1}$ .Now we raise $a^{-1}$ to the power of $-4$ , which is $(a^{-1})^{-4}$ . Using the property $(a^{m})^{n} = a^{m*n}$ , we get $a^{(-1*-4)}$ .
Final Simplification: Simplifying $a^{(2-3)}$ gives us $a^{-1}$ .Now we raise $a^{-1}$ to the power of $-4$ , which is $(a^{-1})^{-4}$ . Using the property $(a^{m})^{n} = a^{m*n}$ , we get $a^{(-1*-4)}$ .Simplifying $a^{(-1*-4)}$ gives us $a^{4}$ .
Final Simplification: Simplifying $a^{(2-3)}$ gives us $a^{-1}$ .Now we raise $a^{-1}$ to the power of $-4$ , which is $(a^{-1})^{-4}$ . Using the property $(a^{m})^{n} = a^{m*n}$ , we get $a^{(-1*-4)}$ .Simplifying $a^{(-1*-4)}$ gives us $a^{4}$ .Now we multiply the result from the first part of the expression, which is $1$ , by the result from the second part of the expression, which is $a^{4}$ . $a^{-1}$ $1$ equals $a^{4}$ .
Final Simplification: Simplifying $a^{(2-3)}$ gives us $a^{-1}$ .Now we raise $a^{-1}$ to the power of $-4$ , which is $(a^{-1})^{-4}$ . Using the property $(a^{m})^{n} = a^{m*n}$ , we get $a^{(-1*-4)}$ . Simplifying $a^{(-1*-4)}$ gives us $a^{4}$ . Now we multiply the result from the first part of the expression, which is $1$ , by the result from the second part of the expression, which is $a^{4}$ . $a^{-1}$ $1$ equals $a^{4}$ . We have now simplified the entire expression to $a^{4}$ , which means that $a^{-1}$ $4$ . Therefore, the value of $a^{-1}$ $5$ is $a^{-1}$ $6$ .