Radicals and Rational Exponents: Apply It 1

  • Calculate square roots and apply them in basic operations like addition, subtraction, and rationalizing the denominators.
  • Use the product and quotient rules to simplify expressions that include square roots.
  • Understand how to use rational exponents in expressions and recognize their connection to roots for simplifying calculations.

Quilt blocks are fabric components sewn together to make a quilt.

Photo by Ibrahim Rifath on Unsplash

The quilt block design of a square within a square is called an “economy quilt block.” A textile artist is making a quilt using economy quilt blocks of varying sizes, so he makes a sketch of the design with side length [latex]w[/latex] as shown below.

The image shows a geometric diagram with two parts. On the left is a square with width "w" containing nested shapes: a purple square rotated 45 degrees (appearing as a diamond) inside a blue square, all set against a green background. Each shape is labeled with measurements showing "w/2" for various dimensions. On the right side is a smaller right triangle in green with sides labeled "w/2", followed by a question mark

The artist needs to calculate how many square inches of fabric he will need for each orange triangle. First, he will need to determine an expression for the hypotenuse.

The image shows a green isosceles triangle with a dashed vertical line down its middle, dividing it into two equal parts. The triangle's base is labeled "base" across the bottom, its height is labeled "height" along the middle dashed line, and both sides are labeled "w/2". A small curved arrow is drawn inside the triangle near the dashed line, indicating an angle.