If [tex]x \neq 0[/tex], what is the sum of [tex]4 \sqrt[3]{x^{10}}+5 x^3 \sqrt[3]{8 x}[/tex] in simplest form?
Answer (1)
Simplify 4 3 x 10 to 4 x 3 3 x .
Simplify 5 x 3 3 8 x to 10 x 3 3 x .
Add the simplified terms: 4 x 3 3 x + 10 x 3 3 x = 14 x 3 3 x .
The final simplified expression is 14 x 3 3 x .
Explanation
Understanding the Problem We are given the expression 4 3 x 10 + 5 x 3 3 8 x , where x = 0 , and we want to simplify it.
Simplifying the First Term First, let's simplify the term 4 3 x 10 . We can rewrite x 10 as x 9 ⋅ x , so we have: 4 3 x 10 = 4 3 x 9 ⋅ x = 4 3 x 9 ⋅ 3 x = 4 x 3 3 x .
Simplifying the Second Term Next, let's simplify the term 5 x 3 3 8 x . We know that 3 8 = 2 , so we have: 5 x 3 3 8 x = 5 x 3 ⋅ 3 8 ⋅ 3 x = 5 x 3 ⋅ 2 ⋅ 3 x = 10 x 3 3 x .
Adding the Simplified Terms Now, we can add the simplified terms: 4 x 3 3 x + 10 x 3 3 x = ( 4 + 10 ) x 3 3 x = 14 x 3 3 x .
Final Answer Therefore, the simplified expression is 14 x 3 3 x .
Examples
Imagine you're calculating the total amount of material needed for a construction project. If you have two different components with volumes expressed in terms of cube roots and powers of a variable, simplifying the expression like this helps you combine those volumes to find the total material required. This kind of simplification is also useful in physics when dealing with quantities that involve roots and exponents, such as calculating energy or momentum.
