Simplify the following expression.
[tex]
\begin{array}{l}
x^3\left(x^5-6 x^2\right) \\
x^{[?]}-\square x
\end{array}
[/tex]
Answer (1)
Distribute x 3 to both terms inside the parentheses: $x^3(x^5 - 6x^2) = x^3
\cdot x^5 - x^3 \cdot (6x^2)$.
Simplify the first term using the rule x a ⋅ x b = x a + b : x 3 ⋅ x 5 = x 3 + 5 = x 8 .
Simplify the second term: x 3 ⋅ ( 6 x 2 ) = 6 ⋅ ( x 3 ⋅ x 2 ) = 6 ⋅ x 3 + 2 = 6 x 5 .
Combine the simplified terms to get the final expression: x 8 − 6 x 5 . The final answer is x 8 − 6 x 5 .
Explanation
Understanding the Problem We are given the expression x 3 ( x 5 − 6 x 2 ) and asked to simplify it. Our goal is to rewrite it in the form x [ ?] − □ x , where we need to find the missing exponent and coefficient.
Distributing First, we distribute x 3 to both terms inside the parentheses:
x 3 ( x 5 − 6 x 2 ) = x 3 ⋅ x 5 − x 3 ⋅ ( 6 x 2 )
Simplifying the First Term Now, we simplify each term separately. For the first term, we use the rule x a ⋅ x b = x a + b :
x 3 ⋅ x 5 = x 3 + 5 = x 8
Simplifying the Second Term For the second term, we also use the rule x a ⋅ x b = x a + b :
x 3 ⋅ ( 6 x 2 ) = 6 ⋅ ( x 3 ⋅ x 2 ) = 6 ⋅ x 3 + 2 = 6 x 5
Combining Terms Now, we combine the simplified terms:
x 8 − 6 x 5
Finding the Missing Values Finally, we compare our simplified expression x 8 − 6 x 5 with the target form x [ ?] − □ x . We see that the exponent of the first term is 8 and the coefficient of the second term is 6. Therefore, the simplified expression is x 8 − 6 x 5 .
Final Answer Thus, the simplified expression is x 8 − 6 x 5 .
Examples
Understanding how to simplify expressions like this is useful in many areas of math and science. For example, if you're calculating the area of a complex shape or modeling the growth of a population, you might end up with an expression that needs to be simplified before you can get a useful answer. Simplifying expressions makes them easier to work with and understand, which is a key skill in problem-solving.
