Radical Expressions with Variables

Worked solutions · Section 4.2, page 205 · Problems 49-55, 57-64

Radical rules

\(n\) = index (the small number on the radical)

1. Product property

\(\sqrt[n]{ab} = \sqrt[n]{a}\cdot\sqrt[n]{b}\)
Same index: split apart or join together

2. Quotient property

\(\sqrt[n]{\dfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}, \quad b \ne 0\)
Root of the top over root of the bottom

3. Odd index

\(\sqrt[n]{x^n} = x \quad (n \text{ odd})\)
Sign stays: \(\sqrt[3]{(-2)^3} = -2\)

4. Even index = absolute value

\(\sqrt[n]{x^n} = |x| \quad (n \text{ even})\)
Never negative: \(\sqrt[4]{(-3)^4} = 3\)
  • No absolute value on even powers: \(\sqrt[4]{(y^2)^4} = y^2\)
  • No absolute value when all variables are positive

5. Pull out perfect powers

\(\sqrt[n]{x^{qn+r}} = x^q\sqrt[n]{x^r}\)
Exponent \(\div\) index: quotient comes out, remainder stays in
\(\sqrt{a^7} = \sqrt{a^6\cdot a} = a^3\sqrt{a}\)
  • Squares: 4, 9, 16, 25, 36, 49, 64, 81
  • Cubes: 8, 27, 64, 125
  • Fourth powers: 16, 81
  • Fifth powers: 32, 243

Exponent rules

6. Multiply powers with the same base (Product rule)

\(a^m\cdot a^n = a^{m+n}\)
Same base: add exponents

7. Divide powers with the same base (Quotient rule)

\(\dfrac{a^m}{a^n} = a^{m-n}\)
Same base: subtract exponents. Watch the signs: \(p^{7-(-1)} = p^8\)

8. Power raised to a power

\((a^m)^n = a^{mn}\)
Multiply exponents. Read backward to spot perfect powers: \(y^8 = (y^2)^4\)

9. Negative exponent = reciprocal

\(a^{-n} = \dfrac{1}{a^n}\)
Moves the factor across the fraction bar

10. Radical = rational exponent

\(\sqrt[n]{a^m} = a^{m/n}\)
Index becomes the denominator. Use it when indices differ

Simplest form

11. Rationalize the denominator

Multiply top and bottom to complete a perfect \(n\)th power
\(\dfrac{1}{\sqrt[5]{n^2}}\cdot\dfrac{\sqrt[5]{n^3}}{\sqrt[5]{n^3}} = \dfrac{\sqrt[5]{n^3}}{\sqrt[5]{n^5}} = \dfrac{\sqrt[5]{n^3}}{n}\)
\(n^2\) needs \(n^3\) more to make \(n^5\)

12. Reduce the index

\(\sqrt[4]{y^2} = y^{2/4} = y^{1/2} = \sqrt{y}\)
Use when the index and exponent share a factor

Problems 49-55: Simplify the expression

Variables may be negative. Even index: absolute value. Odd index: none.

Problem 49

\(\sqrt[4]{81y^8}\)

Rules used:
  • 1. Product property
  • 8. Power raised to a power
  • 4. Even index
Step 1. Rewrite as fourth powers.
\(\sqrt[4]{81y^8} = \sqrt[4]{3^4\cdot (y^2)^4}\)
Step 2. Take each fourth root.
\(\sqrt[4]{3^4}\cdot\sqrt[4]{(y^2)^4} = 3\left|y^2\right|\)
Step 3. \(y^2 \ge 0\), so \(\left|y^2\right| = y^2\).
Exact answer
\(3y^2\)
Problem 50

\(\sqrt[3]{64r^3t^6}\)

Rules used:
  • 1. Product property
  • 8. Power raised to a power
  • 3. Odd index
Step 1. Rewrite as cubes.
\(\sqrt[3]{64r^3t^6} = \sqrt[3]{4^3\cdot r^3\cdot (t^2)^3}\)
Step 2. Take each cube root. Odd index: \(\sqrt[3]{r^3} = r\).
Exact answer
\(4rt^2\)
Problem 51

\(\sqrt[5]{\dfrac{m^{10}}{n^5}}\)

Rules used:
  • 2. Quotient property
  • 8. Power raised to a power
  • 3. Odd index
Step 1. Split top and bottom.
\(\sqrt[5]{\dfrac{m^{10}}{n^5}} = \dfrac{\sqrt[5]{m^{10}}}{\sqrt[5]{n^5}}\)
Step 2. Rewrite as fifth powers.
\(\dfrac{\sqrt[5]{(m^2)^5}}{\sqrt[5]{n^5}}\)
Step 3. Take fifth roots. Odd index: \(\sqrt[5]{n^5} = n\).
Exact answer
\(\dfrac{m^2}{n}\)
Problem 52

\(\sqrt[4]{\dfrac{k^{16}}{16z^4}}\)

Rules used:
  • 2. Quotient property
  • 8. Power raised to a power
  • 4. Even index
Step 1. Split into fourth powers.
\(\sqrt[4]{\dfrac{k^{16}}{16z^4}} = \dfrac{\sqrt[4]{(k^4)^4}}{\sqrt[4]{2^4\cdot z^4}}\)
Step 2. Take fourth roots.
\(\dfrac{\left|k^4\right|}{2|z|} = \dfrac{k^4}{2|z|} \quad (k^4 \ge 0)\)
Exact answer
\(\dfrac{k^4}{2|z|}\)
Problem 53

\(\sqrt[6]{\dfrac{g^6h}{h^7}}\)

Rules used:
  • 7. Quotient rule
  • 2. Quotient property
  • 4. Even index
Step 1. Simplify inside first.
\(\dfrac{h}{h^7} = h^{1-7} = h^{-6} = \dfrac{1}{h^6}, \qquad \sqrt[6]{\dfrac{g^6h}{h^7}} = \sqrt[6]{\dfrac{g^6}{h^6}}\)
Step 2. Split; take sixth roots.
\(\dfrac{\sqrt[6]{g^6}}{\sqrt[6]{h^6}} = \dfrac{|g|}{|h|}\)
Exact answer
\(\dfrac{|g|}{|h|}\)
Also written \(\left|\dfrac{g}{h}\right|\)
Problem 54

\(\sqrt[8]{\dfrac{n^{18}p^7}{n^2p^{-1}}}\)

Rules used:
  • 7. Quotient rule
  • 8. Power raised to a power
  • 4. Even index
Step 1. Simplify inside first.
\(\sqrt[8]{n^{18-2}\,p^{7-(-1)}} = \sqrt[8]{n^{16}p^8}\)
Step 2. Rewrite as eighth powers.
\(\sqrt[8]{(n^2)^8\cdot p^8}\)
Step 3. Take eighth roots.
\(\left|n^2\right|\cdot|p| = n^2|p| \quad (n^2 \ge 0)\)
Exact answer
\(n^2|p|\)
Problem 55 · Error analysis

\(\sqrt[6]{\dfrac{64h^{12}}{g^6}}\)

The book's work
\(\sqrt[6]{\dfrac{64h^{12}}{g^6}} = \dfrac{\sqrt[6]{64h^{12}}}{\sqrt[6]{g^6}} = \dfrac{\sqrt[6]{2^6\cdot(h^2)^6}}{\sqrt[6]{g^6}} = \dfrac{2h^2}{g}\)
Rule used:
  • 4. Even index
Step 1. Find the error.
\(\sqrt[6]{g^6} = |g|, \text{ not } g\)
\(g = -1,\ h = 1: \quad \sqrt[6]{64} = 2, \quad \dfrac{2(1)^2}{-1} = -2\)
Step 2. Correct the last step.
\(\dfrac{\sqrt[6]{2^6\cdot(h^2)^6}}{\sqrt[6]{g^6}} = \dfrac{2h^2}{|g|} \quad (h^2 \ge 0)\)
Exact answer
\(\dfrac{2h^2}{|g|}\)

Problems 57-64: Write in simplest form

All variables are positive. No absolute value needed.

Problem 57

\(\sqrt{81a^7b^{12}c^9}\)

Rules used:
  • 5. Pull out perfect powers
  • 1. Product property
Step 1. Split off perfect squares.
\(\sqrt{81a^7b^{12}c^9} = \sqrt{81a^6b^{12}c^8}\cdot\sqrt{ac}\)
Step 2. Halve each exponent.
\(\sqrt{81a^6b^{12}c^8} = 9a^3b^6c^4\)
Exact answer
\(9a^3b^6c^4\sqrt{ac}\)
Problem 58

\(\sqrt[3]{125r^4s^9t^7}\)

Rules used:
  • 5. Pull out perfect powers
  • 1. Product property
Step 1. Split off perfect cubes.
\(\sqrt[3]{125r^4s^9t^7} = \sqrt[3]{125r^3s^9t^6}\cdot\sqrt[3]{rt}\)
Step 2. Divide exponents by 3.
\(\sqrt[3]{125r^3s^9t^6} = 5rs^3t^2\)
Exact answer
\(5rs^3t^2\sqrt[3]{rt}\)
Problem 59

\(\sqrt[5]{\dfrac{160m^6}{n^7}}\)

Rules used:
  • 2. Quotient property
  • 5. Pull out perfect powers
  • 11. Rationalize the denominator
Step 1. Pull out fifth powers.
\(\dfrac{\sqrt[5]{2^5\cdot 5\cdot m^5\cdot m}}{\sqrt[5]{n^5\cdot n^2}} = \dfrac{2m\sqrt[5]{5m}}{n\sqrt[5]{n^2}}\)
Step 2. Rationalize: \(n^2\cdot n^3 = n^5\).
\(\dfrac{2m\sqrt[5]{5m}}{n\sqrt[5]{n^2}}\cdot\dfrac{\sqrt[5]{n^3}}{\sqrt[5]{n^3}} = \dfrac{2m\sqrt[5]{5mn^3}}{n\sqrt[5]{n^5}} = \dfrac{2m\sqrt[5]{5mn^3}}{n\cdot n}\)
Exact answer
\(\dfrac{2m\sqrt[5]{5mn^3}}{n^2}\)
Problem 60

\(\sqrt[4]{\dfrac{405x^3y^3}{5x^{-1}y}}\)

Rules used:
  • 7. Quotient rule
  • 5. Pull out perfect powers
  • 12. Reduce the index
Step 1. Simplify inside first.
\(\sqrt[4]{\dfrac{405}{5}\,x^{3-(-1)}\,y^{3-1}} = \sqrt[4]{81x^4y^2}\)
Step 2. Pull out fourth powers.
\(\sqrt[4]{3^4\cdot x^4\cdot y^2} = 3x\sqrt[4]{y^2}\)
Step 3. Reduce the index.
\(\sqrt[4]{y^2} = y^{2/4} = y^{1/2} = \sqrt{y}\)
Exact answer
\(3x\sqrt{y}\)
Problem 61

\(\dfrac{\sqrt[3]{w}\cdot\sqrt{w^5}}{\sqrt{25w^{16}}}\)

Rules used:
  • 10. Radical = rational exponent
  • 6. Product rule
  • 7. Quotient rule
Step 1. Simplify the denominator.
\(\sqrt{25w^{16}} = 5w^8\)
Step 2. Switch to rational exponents.
\(\sqrt[3]{w}\cdot\sqrt{w^5} = w^{1/3}\cdot w^{5/2} = w^{2/6 + 15/6} = w^{17/6}\)
Step 3. Split the exponent.
\(w^{17/6} = w^{12/6}\cdot w^{5/6} = w^2\sqrt[6]{w^5}\)
Step 4. Divide.
\(\dfrac{w^2\sqrt[6]{w^5}}{5w^8} = \dfrac{\sqrt[6]{w^5}}{5w^{8-2}}\)
Exact answer
\(\dfrac{\sqrt[6]{w^5}}{5w^6}\)
Problem 62

\(\dfrac{\sqrt[4]{v^6}}{\sqrt[7]{v^5}}\)

Rules used:
  • 10. Radical = rational exponent
  • 7. Quotient rule
Step 1. Switch to rational exponents.
\(\dfrac{\sqrt[4]{v^6}}{\sqrt[7]{v^5}} = \dfrac{v^{6/4}}{v^{5/7}} = \dfrac{v^{3/2}}{v^{5/7}}\)
Step 2. Subtract exponents.
\(v^{3/2 - 5/7} = v^{21/14 - 10/14} = v^{11/14}\)
Step 3. Back to radical form.
Exact answer
\(\sqrt[14]{v^{11}}\)
Also written \(v^{11/14}\)
Problem 63

\(\dfrac{18w^{1/3}v^{5/4}}{27w^{4/3}v^{1/2}}\)

Rules used:
  • 7. Quotient rule
  • 9. Negative exponent
Step 1. Reduce the numbers.
\(\dfrac{18}{27} = \dfrac{2}{3}\)
Step 2. Subtract exponents.
\(w^{1/3-4/3} = w^{-1}, \qquad v^{5/4-2/4} = v^{3/4}\)
Step 3. Make exponents positive.
\(\dfrac{2}{3}\,w^{-1}v^{3/4} = \dfrac{2v^{3/4}}{3w}\)
Exact answer
\(\dfrac{2v^{3/4}}{3w}\)
Problem 64

\(\dfrac{7x^{-3/4}y^{5/2}z^{-2/3}}{56x^{-1/2}y^{1/4}}\)

Rules used:
  • 7. Quotient rule
  • 9. Negative exponent
Step 1. Reduce the numbers.
\(\dfrac{7}{56} = \dfrac{1}{8}\)
Step 2. Subtract exponents.
\(x^{-3/4-(-2/4)} = x^{-1/4}, \qquad y^{10/4-1/4} = y^{9/4}, \qquad z^{-2/3}\)
Step 3. Make exponents positive.
\(\dfrac{1}{8}\,x^{-1/4}y^{9/4}z^{-2/3} = \dfrac{y^{9/4}}{8x^{1/4}z^{2/3}}\)
Exact answer
\(\dfrac{y^{9/4}}{8x^{1/4}z^{2/3}}\)

Quick checklist

  1. Fraction inside the radical? Simplify inside first, then split
  2. Exponent bigger than the index? Quotient comes out, remainder stays in
  3. Even index, variable may be negative? Odd powers that come out get absolute value
  4. Odd index, or all variables positive? No absolute value
  5. Different indices? Rational exponents, common denominator
  6. Radical in the denominator? Multiply to complete a perfect power
  7. Index and exponent share a factor? Reduce the index
  8. Final check? Each base once, positive exponents, no fraction under a radical