Exercise 4.1 — Exponents and Powers
Complete Solutions
Question 1 · Base and Exponent
(a)
\( 2^4 \)
Base = 2, Exponent = 4
(b)
\( (-3)^5 \)
Base = −3, Exponent = 5
(c)
\( \left(\dfrac{3}{4}\right)^5 \)
Base = \( \dfrac{3}{4} \), Exponent = 5
(d)
\( \dfrac{1}{5^4} = 5^{-4} \)
Base = 5, Exponent = −4
(e)
\( \left(\dfrac{-2}{3}\right)^3 \)
Base = \( \dfrac{-2}{3} \), Exponent = 3
Question 2 · Simplify
(a)
\( 3^2 \times \dfrac{1}{3^5} \times (3^2)^6 \)
\( = 3^2 \times 3^{-5} \times 3^{12} \)
\( = 3^{\,2-5+12} \)
\( = 3^{9} \)
= 19683
(b)
\( (-5)^3 \times (-5)^4 \times (-5)^2 \)
\( = (-5)^{\,3+4+2} \)
\( = (-5)^{9} \)
= −1953125
(c)
\( \left(\dfrac{2}{5}\right)^2 \times \left(\dfrac{2}{5}\right)^{-3} \times \left(\dfrac{2}{5}\right)^4 \)
\( = \left(\dfrac{2}{5}\right)^{2-3+4} \)
\( = \left(\dfrac{2}{5}\right)^{3} \)
\( = \dfrac{8}{125} \)
(d)
\( \left[\left(\dfrac{3}{7}\right)^{-1}\right]^{-2} \)
\( = \left(\dfrac{3}{7}\right)^{(-1)\times(-2)} \)
\( = \left(\dfrac{3}{7}\right)^{2} \)
\( = \dfrac{9}{49} \)
Question 3 · Find the Value of x
(a)
\( 5^{x-2} = 25 \)
\( 5^{x-2} = 5^{2} \)
\( x – 2 = 2 \)
x = 4
(b)
\( (2^2)^x = (2^3)^4 \)
\( 2^{2x} = 2^{12} \)
\( 2x = 12 \)
x = 6
(c)
\( 2^x + 2^x + 2^x = 192 \)
\( 3 \times 2^x = 192 \)
\( 2^x = 64 = 2^6 \)
x = 6
(d)
\( 8^{\,2x-5} = 32^{x} \)
\( 8 = 2^3, \quad 32 = 2^5 \)
\( (2^3)^{2x-5} = (2^5)^{x} \)
\( 2^{\,3(2x-5)} = 2^{\,5x} \)
\( 6x – 15 = 5x \)
x = 15
Note: the exponent on 8 was faint in the photo — read here as (2x−5). Tell me if it should be different and I’ll redo this part.
(e)
\( 7^x \times 2^x \times 3^x = 1764 \)
\( (7 \times 2 \times 3)^x = 42^x \)
\( 1764 = 42^2 \)
x = 2
(f)
\( 4\left(\dfrac{10}{x}\right)^2 – 6\left(\dfrac{10}{x}\right)^2 + 3\left(\dfrac{10}{x}\right)^2 = 1 \)
\( (4 – 6 + 3)\left(\dfrac{10}{x}\right)^2 = 1 \)
\( \left(\dfrac{10}{x}\right)^2 = 1 \)
\( \dfrac{10}{x} = \pm 1 \)
x = 10 or x = −10
Question 4 · Evaluate
(a)
\( \left(\dfrac{1}{2}\right)^5 \cdot \left(\dfrac{-2}{3}\right)^4 \cdot \left(\dfrac{3}{5}\right)^{-1} \)
\( = \dfrac{1}{32} \times \dfrac{16}{81} \times \dfrac{5}{3} \)
\( = \dfrac{80}{7776} \)
\( = \dfrac{5}{486} \)
(b)
\( 7^3 \times 49^{-1} \times 5^3 \times 25^{-4} \)
\( 49 = 7^2, \quad 25 = 5^2 \)
\( = 7^3 \times 7^{-2} \times 5^3 \times 5^{-8} \)
\( = 7^{1} \times 5^{-5} \)
\( = \dfrac{7}{3125} \)
(c)
\( \left(\dfrac{12}{5}\right)^3 \times \dfrac{5^6}{144} \)
\( = \dfrac{12^3}{5^3} \times \dfrac{5^6}{144} = \dfrac{12^3 \times 5^3}{144} \)
\( 144 = 12^2 \ \Rightarrow\ \dfrac{12^3}{12^2} = 12 \)
\( = 12 \times 5^3 \)
= 1500
(d)
\( \dfrac{9^3 \times 27 \times 81^4}{3^{-2} \times 3^4 \times 81^2} \)
\( 9 = 3^2, \quad 27 = 3^3, \quad 81 = 3^4 \)
Numerator \( = 3^6 \times 3^3 \times 3^{16} = 3^{25} \)
Denominator \( = 3^{-2} \times 3^4 \times 3^8 = 3^{10} \)
\( = 3^{\,25-10} = 3^{15} \)
= 14,348,907