Weare given the expression:
"The product of five and the square of a number"
So we can write this in math form as:
5 times unknwon number squared
5 * x^2
where we represented the unknown number via the letter "x"
Notice that this letter (called variable in math terms) is squared.
Algebraic expressions that contain a variable to a power different from one, are NOT LINEAR.
This expression in particular that contains the letter (variable) x to the power "2" is referred as a quadratic expression.
Pure mathematical expression is:
We are also asked to represent the following in math terms:
"Multiply five and the reciprocal of the quotient of a number and nine"
in math terms we have a product (given by the order: "multiply"):
5 times the "reciprocal of the quotient of "x" and 9"
We first need to express the quotient they ask for: "a number and 9", representing the "number" with "x" as we did before:
quotient of number and 9 is: x / 9 (x divided by nine)
The "reciprocal" of this is the fraction "flipped-over": 9 / x (nine over x)
Then finally when we complete the expression we get:
5 times (9 / x) = 45/x
In pure math terms:
As we see, in this case , the variable "x" appears in the "denominator" at the very end. Since the variable is in the denominator, the expression is NOT LINEAR. the variable doesn't appear in the numerator with exponent (power) "1" (one) . Having the variable in the denominator involves a "negative exponent" for the variable "x" then this expression is NOT LINEAR.