&(3y/x)^4 [1.2em]
&3*3*3*3* y* y* y * y/x* x * x * x [0.6em]
&81y^4/x^4
Practice makes perfect
a
There are several ways to create equivalent expressions, but we will start by expanding the parentheses. It's raised to the power of 2, meaning it can be written as a product.
(2x-3)^2+5=(2x-3)(2x-3)+5
Let's multiply the binomials using a generic rectangle.
By adding the areas of the smaller rectangles inside the bigger rectangle, we can write an equivalent expression for the square
(2x-3)(2x-3) = 4x^2-6x-6x+9
By substituting this into the original expression, we can rewrite it as several equivalent expressions.
Now we have four expressions that are all equivalent to the first. Let's choose three of them.
(2x-3)(2x-3)+5
4x^2-6x-6x+9+5
4x^2-12x+14
When determining which expression is the simplest, a good general rule is to choose the one with the least number of terms. In this case, that is 4x^2-12x+14.
b
The expression is a fraction raised to the power of 4. That means it can be written as a product of the fraction multiplied four times. But first, notice that both the numerator and denominator contain the factor x^2.
(3x^2y/x^3)^4
Therefore, we can create an equivalent fraction by dividing both the numerator and denominator by x^2.
(3y/x)^4
Now, we can write the power as a product.
All expressions above are equivalent to the first, so let's choose three of them.
(3y/x)^4 [1.5em]
3*3*3*3* y* y* y * y/x* x * x * x [1.25em]
81y^4/x^4
Which one is the simplest is a matter of opinion, but the first and third are the shortest. There's no obvious answer here, so you could argue for both. However, since the last one does not contain parentheses we're going with that.
81y^4/x^4
