25 practice questions on meaning-of-powers 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.
What is the value of $17^1$?
Correct answer: D - $17$
An exponent of $1$ means the base appears exactly once as a factor, so $17^1 = 17$. The value $1$ mistakes the exponent for the answer, $171$ joins the digits, and $34$ doubles the base.
How is $9^2$ read aloud?
Correct answer: C - Nine squared
A power with exponent $2$ is read as the base 'squared', so $9^2$ is 'nine squared' and equals $81$. 'Two squared' would be $2^2 = 4$, 'nine times two' is $18$, and 'nine cubed' is $9^3$.
Expand $4^3$ and find its value.
Correct answer: A - $64$
Expanding gives $4 \times 4 \times 4 = 64$. Multiplying $4$ by $3$ gives $12$, writing the digits side by side gives $43$, and stopping after two factors gives $16$.
Consider: (i) $10^3$ means $10$ used as a factor three times. (ii) $10^3$ means $3$ used as a factor ten times. Which is correct?
Correct answer: B - Only (i)
The exponent counts how many times the base is used, so $10^3 = 10 \times 10 \times 10 = 1000$ and (i) is true. Statement (ii) describes $3^{10}$, which is $59049$, a completely different value. So only (i) survives.
Rewrite $3 \times 3 \times 5 \times 5 \times 5$ using powers.
Correct answer: C - $3^2 \times 5^3$
The $3$ appears twice and the $5$ appears three times, so the product is $3^2 \times 5^3 = 9 \times 125 = 1125$. Swapping the exponents gives $27 \times 25 = 675$. Writing $15^5$ multiplies the bases first, which is wrong, and the last option adds instead of multiplying.
A student claims that $10^4$ and $4^{10}$ mean the same thing because they use the same two numbers. Which reply corrects her?
Correct answer: A - They are read differently, and $10^4 = 10000$ while $4^{10}$ is over a million
In $10^4$ the base is $10$ used four times, giving $10000$, while in $4^{10}$ the base is $4$ used ten times, giving $1048576$. Swapping base and exponent changes both the reading and the value, so they are neither equal nor both $40$, and $10^4$ is the smaller one here.
In the power $7^4$, which number is the exponent?
Correct answer: B - $4$
The small raised number is the exponent, so here it is $4$, and $7$ is the base. The value $28$ comes from multiplying $7$ by $4$ and $11$ from adding them, but neither is the exponent.
Which is greater, $3^4$ or $4^3$, and by how much?
Correct answer: D - $3^4$ is greater by $17$
Expanding gives $3^4 = 81$ and $4^3 = 64$, so $3^4$ wins by $81 - 64 = 17$. The difference is not $7$, and the two are certainly not equal, which shows that swapping base and exponent changes the value.
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