25 practice questions on operations-with-exponents from the Power Play section of the UPSC Prelims syllabus.
25 come with a written explanation.
Try the sample set below - the answer stays hidden until you ask for it.
7 Easy13 Medium5 Hard
Sample questions
Q1
medium
Assertion (A): $(2^2)^3 = 2^6$. Reason (R): When a power is raised to another power, the exponents are multiplied.
ABoth A and R are true, and R is the correct explanation of A
BBoth A and R are true, but R is not the correct explanation of A
CA is true, but R is false
DA is false, but R is true
Show answer and explanation
Correct answer: A - Both A and R are true, and R is the correct explanation of A
A is true, since $(2^2)^3 = 4 \times 4 \times 4 = 64 = 2^6$. R states the rule correctly, and it is exactly why $2 \times 3 = 6$ appears as the exponent. So both are true and R explains A.
Q2
easy
Which single power equals $(2^3)^2$?
A$2^5$
B$2^9$
C$2^6$
D$4^6$
Show answer and explanation
Correct answer: C - $2^6$
When a power is raised to another power the exponents multiply, so $3 \times 2 = 6$ gives $2^6$. Adding them gives $2^5$, raising to the wrong exponent gives $2^9$, and $4^6$ squares the base as well, which the rule does not do.
Q3
medium
Assertion (A): $\frac{2^5 \times 2^3}{2^6} = 2^4$. Reason (R): Exponents are added when powers with the same base are multiplied and subtracted when they are divided.
ABoth A and R are true, and R is the correct explanation of A
BBoth A and R are true, but R is not the correct explanation of A
CA is true, but R is false
DA is false, but R is true
Show answer and explanation
Correct answer: D - A is false, but R is true
R states both rules correctly. Applying them gives $2^{5+3-6} = 2^2$, not $2^4$, so A is false. The assertion would only hold if the bottom exponent were $4$.
Q4
hard
Simplify $\frac{6^7 \times 6^2}{6^5}$.
A$6^4$
B$6^{14}$
C$6^{10}$
D$6^0$
Show answer and explanation
Correct answer: A - $6^4$
First add the exponents on top to get $6^9$, then subtract $5$, leaving $6^4$. Adding all three exponents gives $6^{14}$, and $6^{10}$ comes from subtracting only from the second factor. The answer $6^0$ would need the top and bottom exponents to be equal.
Q5
medium
If $\frac{5^n}{5^3} = 5^4$, what is $n$?
A$7$
B$12$
C$1$
D$4$
Show answer and explanation
Correct answer: A - $7$
The rule gives $n - 3 = 4$, so $n = 7$. Subtracting in the wrong order gives $1$, multiplying the two exponents gives $12$, and $n = 4$ would make the left side $5^1$, not $5^4$.
ABoth A and R are true, and R is the correct explanation of A
BBoth A and R are true, but R is not the correct explanation of A
CA is true, but R is false
DA is false, but R is true
Show answer and explanation
Correct answer: B - Both A and R are true, but R is not the correct explanation of A
A is true because exponents add when the bases match, giving $2^7$. R is true as well, since $2^7$ really is $128$. But knowing the value of $2^7$ does not explain why the exponents add, so R is not the reason behind A.
Q7
medium
Find the value of $(2 \times 5)^3$.
A$30$
B$40$
C$250$
D$1000$
Show answer and explanation
Correct answer: D - $1000$
Since $2 \times 5 = 10$, the value is $10^3 = 1000$, and raising each factor gives the same: $8 \times 125 = 1000$. Raising only the $5$ gives $250$, multiplying all three numbers gives $30$, and $40$ comes from $2^3 \times 5$.
Q8
easy
Which single power equals $(5^2)^4$?
A$25^8$
B$5^{16}$
C$5^6$
D$5^8$
Show answer and explanation
Correct answer: D - $5^8$
Multiplying the exponents gives $2 \times 4 = 8$, so the answer is $5^8$. Adding them gives $5^6$, raising $4$ to the wrong place gives $5^{16}$, and $25^8$ squares the base as well.
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