12 multiple-choice questions, progressively harder.
Simplify 58×553\dfrac{5^8 \times 5}{5^3}5358×5 to a single power of 555.
Solution
Correct answer: C
Add the exponents on top (a plain 555 is 515^151), then subtract the bottom exponent.
58×5153=5953=59−3=56\frac{5^8 \times 5^1}{5^3} = \frac{5^9}{5^3} = 5^{9-3} = 5^65358×51=5359=59−3=56
Fill in the blank: a□a4=a5\dfrac{a^{\square}}{a^4} = a^5a4a□=a5 (with a≠0a \neq 0a=0).
Correct answer: A
The quotient rule subtracts the exponents, so the missing exponent minus 444 is 555.
□−4=5 ⇒ □=9\square - 4 = 5 \;\Rightarrow\; \square = 9□−4=5⇒□=9
Check: a9a4=a5\tfrac{a^9}{a^4} = a^5a4a9=a5.
Simplify (24)3(22)4\dfrac{(2^4)^3}{(2^2)^4}(22)4(24)3 to a single power of 222.
Resolve each power of a power, then subtract the exponents.
(24)3=212,(22)4=28,21228=212−8=24(2^4)^3 = 2^{12}, \qquad (2^2)^4 = 2^8, \qquad \frac{2^{12}}{2^8} = 2^{12-8} = 2^4(24)3=212,(22)4=28,28212=212−8=24
If 7127k=75\dfrac{7^{12}}{7^k} = 7^57k712=75, what is kkk?
The quotient rule subtracts the exponents, so 12−k=512 - k = 512−k=5.
12−k=5 ⇒ k=712 - k = 5 \;\Rightarrow\; k = 712−k=5⇒k=7
Check: 71277=75\tfrac{7^{12}}{7^7} = 7^577712=75.
Simplify 6964×62\dfrac{6^9}{6^4 \times 6^2}64×6269 to a single power of 666.
Correct answer: D
Combine the denominator with the product rule, then subtract.
64×62=66,6966=69−6=636^4 \times 6^2 = 6^6, \qquad \frac{6^9}{6^6} = 6^{9-6} = 6^364×62=66,6669=69−6=63
Simplify (a3b)2\left(\dfrac{a^3}{b}\right)^2(ba3)2 to a single fraction of powers (with b≠0b \neq 0b=0).
A power of a quotient raises the top and bottom separately, and a power of a power multiplies.
(a3b)2=(a3)2b2=a6b2\left(\frac{a^3}{b}\right)^2 = \frac{(a^3)^2}{b^2} = \frac{a^6}{b^2}(ba3)2=b2(a3)2=b2a6
Simplify x5×y6x2×y4\dfrac{x^5 \times y^6}{x^2 \times y^4}x2×y4x5×y6 (with x,y≠0x, y \neq 0x,y=0).
Correct answer: B
Apply the quotient rule to each base on its own, since unlike bases stay separate.
x5x2=x3,y6y4=y2\frac{x^5}{x^2} = x^3, \qquad \frac{y^6}{y^4} = y^2x2x5=x3,y4y6=y2
So the result is x3y2x^3 y^2x3y2.
Which single power of 101010 equals 24×542^4 \times 5^424×54? (Hint: 10=2×510 = 2 \times 510=2×5.)
Both powers share the exponent 444, so use the power-of-a-product rule in reverse.
24×54=(2×5)4=1042^4 \times 5^4 = (2 \times 5)^4 = 10^424×54=(2×5)4=104
Which statement is true?
Use the power-of-a-product rule in reverse: 24×34=(2×3)4=642^4 \times 3^4 = (2 \times 3)^4 = 6^424×34=(2×3)4=64.
24×34=(2×3)4=642^4 \times 3^4 = (2 \times 3)^4 = 6^424×34=(2×3)4=64
The others fail: the base stays 444 (not 161616); a sum is not a single power (24+24=252^4 + 2^4 = 2^524+24=25); and a power of a power multiplies, giving 262^626.
Simplify (2×32)3(2 \times 3^2)^3(2×32)3 to the form 2a×3b2^a \times 3^b2a×3b.
Give the outer exponent to each factor, then resolve the power of a power on 323^232.
(2×32)3=23×(32)3=23×36(2 \times 3^2)^3 = 2^3 \times (3^2)^3 = 2^3 \times 3^6(2×32)3=23×(32)3=23×36
Simplify 23×24+272^3 \times 2^4 + 2^723×24+27 as a single power of 222.
The product rule applies only to the multiplied part; the result is then added to 272^727, and adding two equal powers doubles one of them.
23×24=27,27+27=2×27=282^3 \times 2^4 = 2^7, \qquad 2^7 + 2^7 = 2 \times 2^7 = 2^823×24=27,27+27=2×27=28
Doubling a power of 222 raises its exponent by one.
Simplify (a2b3)2ab2\dfrac{(a^2 b^3)^2}{a b^2}ab2(a2b3)2 (with a,b≠0a, b \neq 0a,b=0).
Apply the outer power to each factor on top, then divide each base separately.
(a2b3)2=a4b6,a4b6a1b2=a4−1b6−2=a3b4(a^2 b^3)^2 = a^4 b^6, \qquad \frac{a^4 b^6}{a^1 b^2} = a^{4-1} b^{6-2} = a^3 b^4(a2b3)2=a4b6,a1b2a4b6=a4−1b6−2=a3b4
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