Mastery Check Solutions

Show What You Know

The area of a figure is determined to be ${(144 x^{7} y^{12} z)}^{\frac{1}{4}}$ square units. Write in simplified radical form.

$144 = 12 \cdot 12 = 2^{4} \cdot 3^{2} {(2^{4} \cdot 3^{2} x^{7} y^{12} z)}^{\frac{1}{4}}$

$2^{\frac{4}{4}} \cdot 3^{\frac{2}{4}} x^{\frac{7}{4}} y^{\frac{12}{4}} z^{\frac{1}{4}}$

$2^{1} \cdot 3^{\frac{2}{4}} x^{1 \frac{3}{4}} y^{3} z^{\frac{1}{4}}$

$2 | x y^{3} | \sqrt[4]{3^{2} x^{3} z}$
$2 | x y^{3} | \sqrt[4]{9 x^{3} z} {units}^{2}$

Note

Remember to write the prime factorization of the coefficient (number) so you can determine the simplified exponent using the index.

Q: Is the index even or odd?

A: Even. 

Q: What do you need to check for when the index is even?

A: That any variable base raised to an odd power is in absolute value bars.

Determine the simplified area of the given rectangle.

$A = l w l = 5 \sqrt[3]{8 x^{2} y} w = \sqrt[3]{2 x^{2} y^{2}}$

$A = (5 \sqrt[3]{8 x^{2} y}) (\sqrt[3]{2 x^{2} y^{2}})$

$A = 5 \sqrt[3]{8 \cdot 2 x^{4} y^{3}}$

$A = 5 \sqrt[3]{2^{4} x^{4} y^{3}} A = 5 (2^{\frac{4}{3}} x^{\frac{4}{3}} y^{\frac{3}{3}}) A = 5 (2^{1 \frac{1}{3}} x^{1 \frac{1}{3}} y^{1})$

$5 \cdot 2 x y \sqrt[3]{2 x}$

$10 x y \sqrt[3]{2 x} {units}^{2}$

Note

Absolute value bars are not needed in the simplified answer because the index is an odd number.

Say What You Know

In your own words, talk about what you have learned using the objectives for this part of the lesson and your work on this page.

Note

Restate the objectives of the lesson in your own words. If you are unable to restate the lesson objectives, go back and reread the objectives and then explain them.

Simplify $d^{th}$ -degree radical expressions. 
Simplify expressions with rational exponents.