Simplify Exponents: Practice Problems & Solutions
A set of practice questions on simplifying exponents is presented below. Detailed explanations covering the rules of exponents are included in the collapsible sections.
Practice Questions
Question 1
\( 27 \left(\frac{1}{9}\right)^2 \left(\frac{9^2}{3^5} \right) = \)
A) 9 B) 18 C) 1/18 D) 27 E) 1/9
Show Detailed Solution
1. Convert everything to base 3: \(27 = 3^3\), \(9 = 3^2\).
2. Substitute: \(3^3 \cdot (1/3^2)^2 \cdot ( (3^2)^2 / 3^5 ) = 3^3 \cdot (1/3^4) \cdot (3^4 / 3^5)\).
3. Apply Product/Quotient Rules: \(3^3 \cdot 3^{-4} \cdot 3^4 \cdot 3^{-5} = 3^{(3 - 4 + 4 - 5)} = 3^{-2} = 1/9\).
Question 2
\( 80 \left(\frac{1}{5^{-1}}\right)^2 \left(\frac{25^{-1}}{4}\right)^2 = \)
A) 1/5 B) 5 C) 25 D) 1/25 E) 16
Show Detailed Solution
1. Simplify: \((1/5^{-1}) = 5\). So, \((5)^2 = 25\).
2. Second term: \(25^{-1} = 1/25\). Thus, \(( (1/25)/4 )^2 = (1/100)^2 = 1/10000\).
3. Calculation: \(80 \cdot 25 \cdot (1/10000) = 2000 / 10000 = 1/5\).
Question 3
\( \left(\frac{x^4}{y^5}\right)^3 \left(\frac{y^2}{x^2}\right)^2 = \)
A) \(\frac{x^6}{y^3}\) B) \(\frac{x^8}{y^{11}}\) C) \(\frac{x^{11}}{y^4}\) D) \(\frac{x^8}{y^3}\) E) \(\frac{x^6}{y^{11}}\)
Show Detailed Solution
1. Apply Power of Power Rule \((a^m)^n = a^{mn}\): \((x^{12} / y^{15}) \cdot (y^4 / x^4)\).
2. Apply Quotient Rule \(a^m/a^n = a^{m-n}\): \(x^{12-4} / y^{15-4} = x^8 / y^{11}\).
Question 4
\( \frac{3(2x + 2y)^5}{4 (x + y)^3} = \)
A) \(\frac{3}{4}(x^2 + y^2)\) B) \(24(x+y)^3\) C) \(24(x+y)^2\) D) \(\frac{3}{2}(x+y)^{5/3}\) E) \(24(x^2 + y^2)\)
Show Detailed Solution
1. Factor numerator: \(3(2(x+y))^5 = 3 \cdot 2^5 \cdot (x+y)^5 = 3 \cdot 32 \cdot (x+y)^5 = 96(x+y)^5\).
2. Divide: \(\frac{96(x+y)^5}{4(x+y)^3} = 24(x+y)^{5-3} = 24(x+y)^2\).
Question 5
\( \left(\frac{x^0}{2y}\right)^2 \left(\frac{y^4}{x^3}\right)^2 = \)
A) \(\frac{y^6}{2x^6}\) B) \(\frac{y^4}{4x^4}\) C) \(\frac{y^4}{4}\) D) \(\frac{y^6}{4x^6}\) E) \(\frac{1}{4x^4}\)
Show Detailed Solution
1. \(x^0 = 1\), so \((1/2y)^2 = 1/(4y^2)\).
2. Second part: \((y^4/x^3)^2 = y^8/x^6\).
3. Combine: \((1/4y^2) \cdot (y^8/x^6) = y^8 / (4x^6y^2) = y^6 / 4x^6\).
Question 6
\( (- 3 x^2 y^3) (- 4 x^3 y^5) = \)
A) \(-12x^5y^8\) B) \(12x^5y^8\) C) \(12x^6y^{15}\) D) \(\frac{9}{64}x^5y^8\) E) \(\frac{9}{64}x^6y^{15}\)
Show Detailed Solution
1. Multiply coefficients: \((-3) \cdot (-4) = 12\).
2. Add exponents for variables: \(x^2 \cdot x^3 = x^{2+3} = x^5\) and \(y^3 \cdot y^5 = y^{3+5} = y^8\).
Result: \(12x^5y^8\).
Question 7
\( \frac{12 x^3 y^{-2}}{4 x^{-2}y^3} = \)
A) \(\frac{3x^5}{y^5}\) B) \(27648x^5y^5\) C) \(48x^5y\) D) \(3xy\) E) \(xy\)
Show Detailed Solution
1. Divide coefficients: \(12/4 = 3\).
2. Subtract exponents: \(x^{3 - (-2)} = x^5\) and \(y^{-2 - 3} = y^{-5}\).
3. Convert negative exponent: \(3x^5y^{-5} = 3x^5/y^5\).
Question 8
\( \left(\frac{3 x^{-2}}{y^2}\right)^{-2} = \)
A) \(9x^4y^4\) B) \(9x^{-4}y^{-4}\) C) \(\frac{x^{-4}y^{-4}}{9}\) D) \(\frac{x^{-4}y^4}{9}\) E) \(\frac{x^4y^4}{9}\)
Show Detailed Solution
1. Apply Power of a Quotient Rule: \((a/b)^{-n} = (b/a)^n\).
2. Invert and square: \((y^2 / 3x^{-2})^2 = (y^4 / 9x^{-4})\).
3. Move \(x^{-4}\) up: \((x^4y^4)/9\).
Question 9
\( \left( \frac{2x^2y^{-1}}{5} \right)^2 \left(\frac{5 x^{-1}y^3}{4}\right)^3 = \)
A) \(5x^4y^7\) B) \(\frac{5}{16}xy\) C) \(\frac{5}{16}xy^7\) D) \(\frac{5}{16}x^4y^7\) E) \(\frac{5}{4}x^4y^7\)
Show Detailed Solution
1. Expand first part: \((4x^4y^{-2})/25\).
2. Expand second part: \((125x^{-3}y^9)/64\).
3. Multiply: \((4/25) \cdot (125/64) \cdot x^{4-3} \cdot y^{-2+9} = (1/1) \cdot (5/16) \cdot x \cdot y^7 = 5/16 xy^7\).
Question 10
\( \left(\frac{x^{-1}}{y^0} \right)^2 \left(\frac{x^2}{y^3}\right)^3 = \)
A) \(\frac{x}{y}\) B) \(\frac{x^2}{y^4}\) C) \(\frac{x^4}{y^6}\) D) \(\frac{x^4}{y^9}\) E) undefined
Show Detailed Solution
1. Note \(y^0=1\). Part 1: \((x^{-1})^2 = x^{-2}\).
2. Part 2: \((x^2/y^3)^3 = x^6/y^9\).
3. Combine: \(x^{-2} \cdot x^6 \cdot y^{-9} = x^4 \cdot y^{-9} = x^4/y^9\).
Question 11
\( \left(\frac{x^2}{y^3}\right) \left(\frac{8x}{y}\right) \left(\frac{x^2}{4}\right)^2 = \)
A) \(\frac{2x^4}{y^4}\) B) \(\frac{x^7}{2y^4}\) C) \(\frac{x^4}{8y^4}\) D) \(\frac{x^4}{y^4}\) E) \(\frac{x^4}{16y^4}\)
Show Detailed Solution
1. Simplify last term: \((x^2/4)^2 = x^4/16\).
2. Expression: \((x^2/y^3) \cdot (8x/y) \cdot (x^4/16) = (8/16) \cdot (x^2 \cdot x \cdot x^4) / (y^3 \cdot y) = (1/2) \cdot x^7 / y^4\).
Result: \(x^7 / (2y^4)\).