Exponential Equations Worksheet 1 - Free Printable

Here are the step-by-step solutions for each exponential equation. The main strategy is to rewrite all numbers as powers of the same base (like 2, 3, 5, etc.) so we can compare the exponents directly.

a) $0.25^{2-x} = \frac{256}{2^{x+3}}$
* Rewrite bases: $0.25 = \frac{1}{4} = 2^{-2}$ and $256 = 2^8$.
* Equation: $(2^{-2})^{2-x} = \frac{2^8}{2^{x+3}}$
* Simplify exponents: $2^{-2(2-x)} = 2^{8-(x+3)}$
* $-4 + 2x = 8 - x - 3$
* $2x + x = 5 + 4 \Rightarrow 3x = 9 \Rightarrow x = 3$

b) $\frac{27^{3x-2}}{243} = 81^{3x-7}$
* Rewrite bases (base 3): $27=3^3$, $243=3^5$, $81=3^4$.
* Equation: $\frac{(3^3)^{3x-2}}{3^5} = (3^4)^{3x-7}$
* Simplify: $\frac{3^{9x-6}}{3^5} = 3^{12x-28}$
* $3^{9x-6-5} = 3^{12x-28} \Rightarrow 9x - 11 = 12x - 28$
* $28 - 11 = 12x - 9x \Rightarrow 17 = 3x \Rightarrow x = \frac{17}{3}$

c) $\frac{1}{5} \times (\frac{1}{625})^{1-x} = 25^{3x+1}$
* Rewrite bases (base 5): $\frac{1}{5}=5^{-1}$, $\frac{1}{625}=5^{-4}$, $25=5^2$.
* Equation: $5^{-1} \times (5^{-4})^{1-x} = (5^2)^{3x+1}$
* Simplify: $5^{-1} \times 5^{-4+4x} = 5^{6x+2}$
* Add exponents on left: $-1 - 4 + 4x = 6x + 2$
* $4x - 5 = 6x + 2 \Rightarrow -7 = 2x \Rightarrow x = -3.5$

d) $2^x \times 5^x = 0.1 \times (10^{x-1})^5$
* Left side: $(2 \times 5)^x = 10^x$. Right side: $10^{-1} \times 10^{5(x-1)} = 10^{-1 + 5x - 5} = 10^{5x-6}$.
* Equation: $10^x = 10^{5x-6}$
* $x = 5x - 6 \Rightarrow 6 = 4x \Rightarrow x = 1.5$

e) $625^{-2x+1} \times \frac{1}{5} = 125^3$
* Rewrite bases (base 5): $625=5^4$, $125=5^3$.
* Equation: $(5^4)^{-2x+1} \times 5^{-1} = (5^3)^3$
* Simplify: $5^{-8x+4} \times 5^{-1} = 5^9$
* $-8x + 4 - 1 = 9 \Rightarrow -8x + 3 = 9$
* $-8x = 6 \Rightarrow x = -\frac{6}{8} = -0.75$

f) $\frac{81^{5-2x} \times 243^{x-2}}{9^{5x-1}} = \frac{1}{3}$
* Rewrite bases (base 3): $81=3^4$, $243=3^5$, $9=3^2$, $\frac{1}{3}=3^{-1}$.
* Numerator: $(3^4)^{5-2x} \times (3^5)^{x-2} = 3^{20-8x} \times 3^{5x-10} = 3^{10-3x}$.
* Denominator: $(3^2)^{5x-1} = 3^{10x-2}$.
* Equation: $\frac{3^{10-3x}}{3^{10x-2}} = 3^{-1}$
* Subtract exponents: $(10 - 3x) - (10x - 2) = -1$
* $12 - 13x = -1 \Rightarrow 13 = 13x \Rightarrow x = 1$

g) $\frac{3^{-9x} \times 3^{-4x+2}}{3^8} = 3^{-5}$
* Combine numerator: $3^{-9x - 4x + 2} = 3^{-13x + 2}$.
* Divide by denominator: $3^{-13x + 2 - 8} = 3^{-13x - 6}$.
* Equation: $-13x - 6 = -5$
* $-13x = 1 \Rightarrow x = -\frac{1}{13}$

h) $\frac{(3^3)^{-3x} \times (3^{-2})^{2x-1}}{(3^4)^2} = 3^{-5}$
* Simplify terms: Num: $3^{-9x} \times 3^{-4x+2} = 3^{-13x+2}$. Denom: $3^8$.
* Equation: $\frac{3^{-13x+2}}{3^8} = 3^{-5}$
* Exponents: $(-13x + 2) - 8 = -5$
* $-13x - 6 = -5 \Rightarrow -13x = 1 \Rightarrow x = -\frac{1}{13}$

i) $\frac{10^{\frac{1}{3}} \times 1000}{10^x} = 0.01$
* Rewrite: $1000=10^3$, $0.01=10^{-2}$.
* Numerator: $10^{1/3} \times 10^3 = 10^{10/3}$.
* Equation: $\frac{10^{10/3}}{10^x} = 10^{-2}$
* Exponents: $\frac{10}{3} - x = -2$
* $x = \frac{10}{3} + 2 = \frac{10}{3} + \frac{6}{3} = \frac{16}{3}$

j) $2 \times 4^{6x} + 64 \times 64^{2x-1} = 39$
* Rewrite bases (base 2 or 4): Let's use base 4. $2=\sqrt{4}=4^{0.5}$? No, let's use base 2.
* $2 \times (2^2)^{6x} + 2^6 \times (2^6)^{2x-1} = 39$
* $2 \times 2^{12x} + 2^6 \times 2^{12x-6} = 39$
* $2^{12x+1} + 2^{12x} = 39$
* Factor out $2^{12x}$: $2^{12x}(2^1 + 1) = 39$
* $2^{12x}(3) = 39 \Rightarrow 2^{12x} = 13$
* $12x = \log_2(13) \Rightarrow x = \frac{\log_2(13)}{12}$

k) $\frac{16^{-x} \times (\frac{1}{256})^{-x-3}}{64^{2x}} = 2048$
* Rewrite bases (base 2): $16=2^4$, $256=2^8$, $64=2^6$, $2048=2^{11}$.
* Term 1: $(2^4)^{-x} = 2^{-4x}$.
* Term 2: $(2^{-8})^{-x-3} = 2^{8x+24}$.
* Denom: $(2^6)^{2x} = 2^{12x}$.
* Equation: $\frac{2^{-4x} \times 2^{8x+24}}{2^{12x}} = 2^{11}$
* Exponents: $(-4x + 8x + 24) - 12x = 11$
* $-8x + 24 = 11 \Rightarrow -8x = -13 \Rightarrow x = \frac{13}{8} = 1.625$

l) $\frac{625^{1-x} \times 25^{x-1}}{3125^x} = 5 \times \frac{1}{5^{6x}}$
* Rewrite bases (base 5): $625=5^4$, $25=5^2$, $3125=5^5$.
* Num: $(5^4)^{1-x} \times (5^2)^{x-1} = 5^{4-4x} \times 5^{2x-2} = 5^{2-2x}$.
* Denom: $5^{5x}$.
* RHS: $5^1 \times 5^{-6x} = 5^{1-6x}$.
* Equation: $\frac{5^{2-2x}}{5^{5x}} = 5^{1-6x}$
* Exponents: $(2 - 2x) - 5x = 1 - 6x$
* $2 - 7x = 1 - 6x \Rightarrow 1 = x \Rightarrow x = 1$

m) $\frac{5^{4-4x} \times 5^{5x-2}}{5^{5x}} = 5^1 \times 5^{-6x}$
* Num: $5^{4-4x+5x-2} = 5^{x+2}$.
* LHS: $\frac{5^{x+2}}{5^{5x}} = 5^{x+2-5x} = 5^{2-4x}$.
* RHS: $5^{1-6x}$.
* Equation: $2 - 4x = 1 - 6x$
* $2x = -1 \Rightarrow x = -0.5$

n) $15^{2x} - 3 \times 225^x = -30$
* Note that $225 = 15^2$. So $225^x = (15^2)^x = 15^{2x}$.
* Equation: $15^{2x} - 3(15^{2x}) = -30$
* $-2(15^{2x}) = -30 \Rightarrow 15^{2x} = 15$
* $2x = 1 \Rightarrow x = 0.5$

o) $\frac{16^{x-1} \times 2^{5x+2}}{32^{2-3x}} = 128$
* Rewrite bases (base 2): $16=2^4$, $32=2^5$, $128=2^7$.
* Num: $(2^4)^{x-1} \times 2^{5x+2} = 2^{4x-4} \times 2^{5x+2} = 2^{9x-2}$.
* Denom: $(2^5)^{2-3x} = 2^{10-15x}$.
* Equation: $\frac{2^{9x-2}}{2^{10-15x}} = 2^7$
* Exponents: $(9x - 2) - (10 - 15x) = 7$
* $24x - 12 = 7 \Rightarrow 24x = 19 \Rightarrow x = \frac{19}{24}$

p) $\frac{32^{1-x} \times 16^{2-2x}}{128^{3x}} = 4096$
* Rewrite bases (base 2): $32=2^5$, $16=2^4$, $128=2^7$, $4096=2^{12}$.
* Num: $(2^5)^{1-x} \times (2^4)^{2-2x} = 2^{5-5x} \times 2^{8-8x} = 2^{13-13x}$.
* Denom: $(2^7)^{3x} = 2^{21x}$.
* Equation: $\frac{2^{13-13x}}{2^{21x}} = 2^{12}$
* Exponents: $13 - 13x - 21x = 12$
* $13 - 34x = 12 \Rightarrow 1 = 34x \Rightarrow x = \frac{1}{34}$

q) $5^{2x} + 125 \times 25^x = 252$
* Rewrite bases (base 5): $125=5^3$, $25=5^2$.
* Equation: $5^{2x} + 5^3 \times (5^2)^x = 252$
* $5^{2x} + 5^3 \times 5^{2x} = 252$
* Factor $5^{2x}$: $5^{2x}(1 + 125) = 252$
* $126 \times 5^{2x} = 252 \Rightarrow 5^{2x} = 2$
* $2x = \log_5(2) \Rightarrow x = \frac{\log_5(2)}{2}$

r) $2x + \sqrt[3]{64} = 6 - 2x\sqrt[3]{128^2}$
* Simplify roots: $\sqrt[3]{64} = 4$. $\sqrt[3]{128^2} = \sqrt[3]{(2^7)^2} = \sqrt[3]{2^{14}} = 2^{14/3} = 2^4 \cdot 2^{2/3} = 16\sqrt[3]{4}$.
* Equation: $2x + 4 = 6 - 2x(16\sqrt[3]{4})$
* $2x + 32x\sqrt[3]{4} = 2$
* $2x(1 + 16\sqrt[3]{4}) = 2$
* $x(1 + 16\sqrt[3]{4}) = 1 \Rightarrow x = \frac{1}{1 + 16\sqrt[3]{4}}$

s) $2 \times 4^{4x} + 32 \times 16^{2x-1} = 1024$
* Rewrite bases (base 2): $4=2^2$, $32=2^5$, $16=2^4$, $1024=2^{10}$.
* Term 1: $2 \times (2^2)^{4x} = 2 \times 2^{8x} = 2^{8x+1}$.
* Term 2: $2^5 \times (2^4)^{2x-1} = 2^5 \times 2^{8x-4} = 2^{8x+1}$.
* Equation: $2^{8x+1} + 2^{8x+1} = 2^{10}$
* $2 \times 2^{8x+1} = 2^{10} \Rightarrow 2^{8x+2} = 2^{10}$
* $8x + 2 = 10 \Rightarrow 8x = 8 \Rightarrow x = 1$

t) $4^{x+3} \times 8^{x-3} = 128$
* Rewrite bases (base 2): $4=2^2$, $8=2^3$, $128=2^7$.
* Equation: $(2^2)^{x+3} \times (2^3)^{x-3} = 2^7$
* $2^{2x+6} \times 2^{3x-9} = 2^7$
* Exponents: $(2x + 6) + (3x - 9) = 7$
* $5x - 3 = 7 \Rightarrow 5x = 10 \Rightarrow x = 2$

u) $\frac{32^{x-1} 512^{1-2x} (4^{-1})^{-3x} (0.125^2)^{x+2}}{256^{-x+3} 8^{5-5x}} = 4096$
* This is a long calculation using base 2.
* $32=2^5, 512=2^9, 4^{-1}=2^{-2}, 0.125=2^{-3}, 256=2^8, 8=2^3, 4096=2^{12}$.
* Numerator exponents sum:
* $5(x-1) = 5x-5$
* $9(1-2x) = 9-18x$
* $-2(-3x) = 6x$
* $2(-3)(x+2) = -6(x+2) = -6x-12$
* Sum Num: $5x - 5 + 9 - 18x + 6x - 6x - 12 = -13x - 8$
* Denominator exponents sum:
* $8(-x+3) = -8x+24$
* $3(5-5x) = 15-15x$
* Sum Denom: $-8x + 24 + 15 - 15x = -23x + 39$
* Total Exponent: Num - Denom = $(-13x - 8) - (-23x + 39) = 10x - 47$.
* Equation: $2^{10x - 47} = 2^{12}$
* $10x - 47 = 12 \Rightarrow 10x = 59 \Rightarrow x = 5.9$

v) $\frac{27^{-3x} \times (\frac{1}{9})^{2x-1}}{81^2} = \frac{1}{243}$
* Rewrite bases (base 3): $27=3^3$, $9=3^2$, $81=3^4$, $243=3^5$.
* Num: $(3^3)^{-3x} \times (3^{-2})^{2x-1} = 3^{-9x} \times 3^{-4x+2} = 3^{-13x+2}$.
* Denom: $(3^4)^2 = 3^8$.
* RHS: $3^{-5}$.
* Equation: $\frac{3^{-13x+2}}{3^8} = 3^{-5}$
* Exponents: $(-13x + 2) - 8 = -5$
* $-13x - 6 = -5 \Rightarrow -13x = 1 \Rightarrow x = -\frac{1}{13}$

Final Answer:
a) $x = 3$
b) $x = \frac{17}{3}$
c) $x = -3.5$
d) $x = 1.5$
e) $x = -0.75$
f) $x = 1$
g) $x = -\frac{1}{13}$
h) $x = -\frac{1}{13}$
i) $x = \frac{16}{3}$
j) $x = \frac{\log_2(13)}{12}$
k) $x = 1.625$
l) $x = 1$
m) $x = -0.5$
n) $x = 0.5$
o) $x = \frac{19}{24}$
p) $x = \frac{1}{34}$
q) $x = \frac{\log_5(2)}{2}$
r) $x = \frac{1}{1 + 16\sqrt[3]{4}}$
s) $x = 1$
t) $x = 2$
u) $x = 5.9$
v) $x = -\frac{1}{13}$