Exercise 4.2 — Exponents and Powers
Complete Solutions
Question 1 · Express in Exponential Form
Rule used: \( \sqrt[n]{a^{m}} = a^{m/n} \)
(a)\( \sqrt[7]{100} \)
\( = 100^{1/7} \)
(b)\( \sqrt[4]{34} \)
\( = 34^{1/4} \)
(c)\( \sqrt[11]{25^{2}} \)
\( = 25^{2/11} \)
(d)\( \sqrt[7]{29^{2}} \)
\( = 29^{2/7} \)
(e)\( \sqrt[7]{\dfrac{10}{7}} \)
\( = \left(\dfrac{10}{7}\right)^{1/7} \)
(f)\( \sqrt[12]{111^{3}} \)
\( = 111^{3/12} = 111^{1/4} \)
(g)\( \sqrt[3]{2^{-6}} \)
\( = 2^{-6/3} = 2^{-2} \)
(h)\( \sqrt[4]{\left(\dfrac{7}{8}\right)^{-5}} \)
\( = \left(\dfrac{7}{8}\right)^{-5/4} \)
Question 2 · Express as Radicals
Rule used: \( a^{m/n} = \sqrt[n]{a^{m}} \)
(a)\( 7^{1/5} \)
\( = \sqrt[5]{7} \)
(b)\( 21^{1/8} \)
\( = \sqrt[8]{21} \)
(c)\( (29)^{2/3} \)
\( = \sqrt[3]{29^{2}} \)
(d)\( (335)^{7/5} \)
\( = \sqrt[5]{335^{7}} \)
(e)\( \left(\dfrac{8}{15}\right)^{1/9} \)
\( = \sqrt[9]{\dfrac{8}{15}} \)
(f)\( \left(\dfrac{17}{327}\right)^{3/7} \)
\( = \sqrt[7]{\left(\dfrac{17}{327}\right)^{3}} \)
(g)\( \left(\dfrac{17}{21}\right)^{2/7} \)
\( = \sqrt[7]{\left(\dfrac{17}{21}\right)^{2}} \)
(h)\( \left(\dfrac{1}{16}\right)^{1/5} \)
\( = \sqrt[5]{\dfrac{1}{16}} \)
Question 3 · Express with Positive Indices
(a)\( x^{-4} \)
\( = \dfrac{1}{x^{4}} \)
(b)\( x^{-1/3} \)
\( = \dfrac{1}{x^{1/3}} \)
(c)\( x^{-2/5} \)
\( = \dfrac{1}{x^{2/5}} \)
(d)\( \dfrac{3}{5}x^{-5/8} \)
\( = \dfrac{3}{5\,x^{5/8}} \)
(e)\( (x^{-4})^{3} \)
\( = \dfrac{1}{x^{12}} \)
(f)\( x^{-5} \times x^{-7} \)
\( = x^{-5-7} = x^{-12} \)
\( = \dfrac{1}{x^{12}} \)
(g)\( \left[\left(\dfrac{1}{x}\right)^{-1/2}\right]^{-2/3} \)
\( = \left(\dfrac{1}{x}\right)^{(-1/2)\times(-2/3)} = \left(\dfrac{1}{x}\right)^{1/3} \)
\( = \dfrac{1}{x^{1/3}} \)
(h)\( 3x^{2/5} \times 3x^{-12/5} \)
\( = 9\,x^{\,2/5\,-\,12/5} = 9x^{-2} \)
\( = \dfrac{9}{x^{2}} \)
(i)\( (3x^{2})^{0} \times \dfrac{1}{(3x^{2})^{-1}} \times 4x^{2} \)
\( (3x^2)^0 = 1, \quad \dfrac{1}{(3x^2)^{-1}} = 3x^2 \)
\( = 1 \times 3x^2 \times 4x^2 \)
\( = 12x^{4} \)
Question 4 · Evaluate
(a)\( 8^{2/3} \)
\( 8 = 2^3 \Rightarrow (2^3)^{2/3} = 2^2 \)
= 4
(b)\( 27^{-2/3} \)
\( 27 = 3^3 \Rightarrow (3^3)^{-2/3} = 3^{-2} \)
\( = \dfrac{1}{9} \)
(c)\( \dfrac{1}{81^{-3/4}} = 81^{3/4} \)
\( 81 = 3^4 \Rightarrow (3^4)^{3/4} = 3^3 \)
= 27
(d)\( 27^{1/3} \times 16^{-1/4} \)
\( 27^{1/3} = 3, \quad 16^{-1/4} = 2^{-1} = \dfrac{1}{2} \)
\( = \dfrac{3}{2} \)
(e)\( (125)^{-2/3} \)
\( 125 = 5^3 \Rightarrow (5^3)^{-2/3} = 5^{-2} \)
\( = \dfrac{1}{25} \)
(f)\( (256)^{-3/8} \)
\( 256 = 2^8 \Rightarrow (2^8)^{-3/8} = 2^{-3} \)
\( = \dfrac{1}{8} \)
(g)\( \left[(625)^{3/4}\right]^{-4/3} \)
\( = 625^{(3/4)\times(-4/3)} = 625^{-1} \)
\( = \dfrac{1}{625} \)
(h)\( 25^{3/2} \times 16^{-3/4} \times 5^{-2} \times 1024^{3/2} \)
\( 25^{3/2}=5^3,\ 16^{-3/4}=2^{-3},\ 1024^{3/2}=2^{15} \)
\( = 5^{3-2} \times 2^{-3+15} = 5^1 \times 2^{12} \)
= 20,480
(i)\( \dfrac{1}{\left[(3^{4})^{1/2}\right]^{-2}} \)
\( (3^4)^{1/2} = 3^2 = 9,\quad 9^{-2} = \dfrac{1}{81} \)
= 81
(j)\( \dfrac{4^{3} \times 16^{-1/4} \times 1}{2^{6}} \times \dfrac{1}{2^{-7}} \)
\( 4^3=2^6,\ 16^{-1/4}=2^{-1} \Rightarrow \dfrac{2^6\times2^{-1}}{2^6}=2^{-1} \)
\( 2^{-1}\times2^{7} = 2^{6} \)
= 64
(k)\( (125)^{-2/3} \times (64)^{4/3} \)
\( = 5^{-2} \times 2^{8\times4/3} = \dfrac{1}{25} \times 256 \)
\( = \dfrac{256}{25} \)
(l)\( (32)^{-2/5} \div (125)^{-2/3} \)
\( 32^{-2/5}=2^{-2}=\dfrac{1}{4},\quad 125^{-2/3}=5^{-2}=\dfrac{1}{25} \)
\( \dfrac{1}{4}\div\dfrac{1}{25} = \dfrac{1}{4}\times25 \)
\( = \dfrac{25}{4} \)
(m)\( 4 \times 81^{-1/2} \times \left(81^{1/2} + 81^{3/2}\right) \)
\( 81^{1/2}=9,\quad 81^{3/2}=729,\quad 81^{-1/2}=\dfrac{1}{9} \)
\( 4\times\dfrac{1}{9}\times(9+729)=4\times\dfrac{738}{9} = 4\times82 \)
= 328
(n)\( (256)^{-(4-3/2)} \)
Exponent \( = -\left(4-\dfrac{3}{2}\right) = -\dfrac{5}{2} \)
\( 256 = 2^8 \Rightarrow 2^{8\times(-5/2)} = 2^{-20} \)
\( = \dfrac{1}{1048576} \)
(o)\( \dfrac{36^{7/2} – 36^{9/2}}{36^{5/2}} \)
\( 36^{7/2}=36^{5/2}\times36,\quad 36^{9/2}=36^{5/2}\times36^2 \)
\( = \dfrac{36^{5/2}(36-1296)}{36^{5/2}} = 36-1296 \)
= −1260
(p)\( \dfrac{(64)^{-1/6}\times(216)^{-1/3}\times(81)^{1/4}}{(512)^{-1/3}\times(16)^{1/4}\times(9)^{-1/2}} \)
Numerator \( = 2^{-1}\times(2^{-1}3^{-1})\times3 = \dfrac{1}{4} \)
Denominator \( = 2^{-3}\times2\times3^{-1} = \dfrac{1}{12} \)
\( \dfrac{1/4}{1/12} \)
= 3
(q)\( \left(\dfrac{256}{6561}\right)^{-5/8} \)
\( \dfrac{256}{6561} = \left(\dfrac{2}{3}\right)^{8} \)
\( \left[\left(\dfrac{2}{3}\right)^{8}\right]^{-5/8} = \left(\dfrac{2}{3}\right)^{-5} = \left(\dfrac{3}{2}\right)^{5} \)
\( = \dfrac{243}{32} \)
(r)\( \left(\dfrac{117649}{1771561}\right)^{1/6} \)
\( 117649 = 7^6,\quad 1771561 = 11^6 \)
\( \left[\left(\dfrac{7}{11}\right)^6\right]^{1/6} \)
\( = \dfrac{7}{11} \)
(s)\( (5^{2}+12^{2})^{1/2} \)
\( = (25+144)^{1/2} = 169^{1/2} \)
= 13
(t)\( (0.04)^{3/2} \)
\( 0.04 = \left(\dfrac{1}{5}\right)^2 \Rightarrow \left(\dfrac{1}{5}\right)^{2\times3/2} = \left(\dfrac{1}{5}\right)^3 \)
\( = \dfrac{1}{125} = 0.008 \)
(u)\( (0.008)^{2/3} \)
\( 0.008 = \left(\dfrac{1}{5}\right)^3 \Rightarrow \left(\dfrac{1}{5}\right)^{3\times2/3} = \left(\dfrac{1}{5}\right)^2 \)
\( = \dfrac{1}{25} = 0.04 \)
Note: the exponent’s denominator looked like a faint 9 in the photo — read here as 2/3, which pairs cleanly with (t) and (v). Tell me if it’s really 2/9 and I’ll redo it.
(v)\( (0.000064)^{5/6} \)
\( 0.000064 = \left(\dfrac{1}{5}\right)^6 \Rightarrow \left(\dfrac{1}{5}\right)^{6\times5/6} = \left(\dfrac{1}{5}\right)^5 \)
\( = \dfrac{1}{3125} = 0.00032 \)
Question 5 · Simplify
\( \left\{\sqrt[3]{x^{4}y} \times \dfrac{1}{\sqrt[4]{x^{2}y^{8}}}\right\}^{-6} \)
\( \sqrt[3]{x^4y} = x^{4/3}y^{1/3}, \qquad \dfrac{1}{\sqrt[4]{x^2y^8}} = x^{-1/2}y^{-2} \)
Inside braces: \( x^{4/3-1/2}\,y^{1/3-2} = x^{5/6}\,y^{-5/3} \)
Raising to \( -6 \): \( x^{(5/6)\times(-6)}\,y^{(-5/3)\times(-6)} = x^{-5}y^{10} \)
\( = \dfrac{y^{10}}{x^{5}} \)
Question 6 · Simplify
\( \sqrt[4]{x^{4}y^{4}} \div \sqrt[3]{x^{3}y^{3}} \)
\( \sqrt[4]{x^4y^4} = xy, \qquad \sqrt[3]{x^3y^3} = xy \)
\( = \dfrac{xy}{xy} = 1 \)
Question 7 · Determine x
(a)
\( 3^{1/5} \times 3^{1/6} = 3^{-x} \)
\( 3^{\,1/5+1/6} = 3^{\,6/30+5/30} = 3^{11/30} \)
\( -x = \dfrac{11}{30} \)
\( x = -\dfrac{11}{30} \)
(b)
\( 800 = 8 \times 10^{8} \times x^{-3/2} \)
\( x^{-3/2} = \dfrac{800}{8\times10^8} = 10^{-6} \)
\( x^{3/2} = 10^{6} \ \Rightarrow\ x = (10^6)^{2/3} = 10^4 \)
x = 10,000
Question 8
By what number should we multiply \( 81^{3/16} \) so that the product becomes \( 3^{5/4} \)?
\( 81^{3/16} = (3^4)^{3/16} = 3^{3/4} \)
Required number \( k \): \( 3^{3/4}\times k = 3^{5/4} \ \Rightarrow\ k = 3^{5/4-3/4} = 3^{1/2} \)
\( k = \sqrt{3} \)
Question 9
By what number should we divide \( (121)^{7/8} \) to obtain \( (1331)^{3/4} \)?
\( (121)^{7/8} = (11^2)^{7/8} = 11^{7/4}, \qquad (1331)^{3/4} = (11^3)^{3/4} = 11^{9/4} \)
Required number \( k \): \( \dfrac{11^{7/4}}{k} = 11^{9/4} \ \Rightarrow\ k = 11^{7/4-9/4} = 11^{-1/2} \)
\( k = \dfrac{1}{\sqrt{11}} \) (i.e. dividing by \( \tfrac{1}{\sqrt{11}} \) is the same as multiplying by \( \sqrt{11} \))
Question 10
If \( x = \dfrac{2}{4} \), find the value of \( (2x)^{x} \).
\( x = \dfrac{2}{4} = \dfrac{1}{2} \)
\( 2x = 2\times\dfrac{1}{2} = 1 \)
\( (2x)^x = 1^{1/2} \)
= 1
Note: the photo showed x as a small stacked fraction that read like “24” — taken here as 2/4 = 1/2, which gives this clean result. Tell me the exact value if it was meant differently.
Question 11
Determine x and y so that \( 3^{x+y} = 81 \) and \( 81^{x-y} = 3 \).
\( 3^{x+y} = 3^4 \ \Rightarrow\ x+y = 4 \)
\( 81^{x-y} = 3 \ \Rightarrow\ (3^4)^{x-y}=3^1 \ \Rightarrow\ 4(x-y)=1 \ \Rightarrow\ x-y = \dfrac{1}{4} \)
Adding: \( 2x = 4+\dfrac{1}{4} = \dfrac{17}{4} \ \Rightarrow\ x = \dfrac{17}{8} \)
Subtracting: \( 2y = 4-\dfrac{1}{4} = \dfrac{15}{4} \ \Rightarrow\ y = \dfrac{15}{8} \)
\( x = \dfrac{17}{8}, \quad y = \dfrac{15}{8} \)