Largest Expression
Which of these five algebraic expressions is largest, given $x$ is between 0 and 1?
Age
14 to 16
Challenge level
Problem
Given that $x$ is positive and less than $1$, list the following five expressions from smallest to largest:
$x^2 + x$ $x^2$ $x^3$ $x^3 + x^2$ $x^4$
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: $x^4 \lt x^3 \lt x^2 \lt x^3 + x^2 \lt x^2 + x$
$x^2 + x \gt x^2$ because $x$ is positive
$x^3 = x^2\times x \lt x^2$ because $x\lt1$
$x^3 + x^2 \gt x^3, x^2$ because $x^3$ and $x^2$ are positive
$x^3 + x^2 = x\times\left(x^2 + x\right) \lt x^2 + x$ (or $x^3 + x^2 = x^2 + x^3 \lt x^2 + x$ because $x^3\lt x$)
And $x^4 =x\times x^3 \lt x^3$
So $x^4 \lt x^3 \lt x^2 \lt x^3 + x^2 \lt x^2 + x$