Operations on real numbers and the laws of exponents

Correct answer: A. (3^7)

Explanation: When the base is the same, exponents are added, so (3^2\cdot3^5=3^7). Add exponents when bases are equal.

Correct answer: B. (8^3)

Explanation: When the base is the same, exponents are subtracted, so (\frac{8^7}{8^4}=8^3). Subtract exponents for same-base division.

Correct answer: B. 4⁶

Explanation: The governing rule is the power-of-a-power law: (aᵐ)ⁿ=aᵐⁿ. The outer exponent multiplies the inner exponent, so (4³)²=4^(3×2)=4⁶. Therefore option B is correct. The exponents are multiplied, not added; consequently 4⁵ is not obtained from this expression. The value 4⁹ would result from another incorrect combination of the exponents. Option D, 4³, ignores the outer square and therefore also fails to preserve the value. The same result can be checked numerically: 4³=64 and 64²=4096, while 4⁶=4096. The revised fourth option is deliberately 4³ rather than an equivalent expression such as 16³, ensuring that only one listed option represents the intended simplified form.

Correct answer: B. 1

Explanation: By the laws of exponents, the zero power of every non-zero number is 1: \(b^0=1\), provided \(b\neq0\). The option \(b\) is incorrect because it represents \(b^1\), not \(b^0\). Exam tip: whenever the base is non-zero, its zero power is always 1.

Correct answer: A. (\frac{1}{8})

Explanation: A negative exponent moves the base to the denominator, so (2^{-3}=\frac{1}{2^3}=\frac{1}{8}). A negative exponent does not mean a negative answer.

Correct answer: A. (3^2\cdot4^2)

Explanation: The power of a product applies to every factor, so ((3\cdot4)^2=3^2\cdot4^2). Apply the exponent to all factors inside the bracket.

Correct answer: B. \(\frac{25}{36}\)

Explanation: The power of a fraction applies to both numerator and denominator. Therefore \(\left(\frac{5}{6}\right)^2=\frac{25}{36}\).

Correct answer: A. \(11^6\)

Explanation: When powers with the same base are multiplied, their exponents are added: \(a^m \cdot a^n = a^{m+n}\). Therefore, \(11^4 \cdot 11^2 = 11^{4+2} = 11^6\). Option B uses an incorrect exponent, while option C represents subtraction of exponents. Remember to add exponents when multiplying powers with the same base.

Correct answer: A. \(6^4\)

Explanation: When powers with the same base are divided, their exponents are subtracted: \(a^m \div a^n=a^{m-n}\). Thus, \(6^9\div6^5=6^{9-5}=6^4\), so option A is correct. Option B results from adding the exponents, which is used for multiplication, not division. Exam tip: for division with the same base, subtract the denominator’s exponent from the numerator’s exponent.

Correct answer: C. 13

Explanation: According to the law of exponents, the first power of any non-zero number is the number itself: \(a^1=a\). Therefore, \(13^1=13\). Option D, 169, is \(13^2\), not \(13^1\). Exam tip: a power of 1 does not change the base.

Correct answer: B. 10

Explanation: By the zero-exponent law,
a^0=1
for every non-zero number, so
9^0=1
. Also,
3^2=9
. Therefore,
(9^0+3^2)=1+9=10
. Exam tip: evaluate exponents before carrying out addition or any other operation.

Correct answer: C. 33

Explanation: Evaluate the powers first: ^2=25 and 2^3=8. Adding them gives 25+8=33, so option C is correct. Option B is only the value of ^2, while option D may result from handling the powers incorrectly. Exam tip: calculate exponents before performing addition or subtraction.

Correct answer: A. (c^m\cdot c^n=c^{m+n})

Explanation: When powers with the same base are multiplied, the exponent tells how many copies of that base are present. Thus \(c^m\) contains \(m\) copies of \(c\), and \(c^n\) contains \(n\) more copies. Multiplying them gives a total of \(m+n\) copies, which is written as \(c^{m+n}\). This is the product rule for exponents.

Therefore \(c^m\cdot c^n=c^{m+n}\), provided the expression is defined under the usual exponent conditions. The exponent is not multiplied, subtracted, or divided in this situation. Subtraction is used when dividing like bases, while multiplication of exponents appears in a power raised to another power, such as \((c^m)^n\). Hence option A is the correct law.

Correct answer: B. d^(r-s)

Explanation: The governing concept is the quotient law of exponents. When nonzero powers with the same base are divided, the exponent in the denominator is subtracted from the exponent in the numerator: d^r/d^s = d^(r−s), provided d≠0. Thus option B is correct. For example, if r=5 and s=2, then d^5/d^2 = d^(5−2)=d^3, which confirms the rule. Option A incorrectly adds the exponents, a rule associated with multiplication of like bases. Option C incorrectly multiplies the exponents, which belongs to raising a power to another power. Option D reverses the subtraction order and generally gives the reciprocal result, so it is not correct.

Correct answer: B.
z^{12}

Explanation: By the power-of-a-power rule,
(a^m)^n=a^{mn}
. Hence,
(z^4)^3=z^{4\cdot3}=z^{12}
. Option A incorrectly adds the exponents, while option C treats the exponent as
4^3
. Exam tip: when a power is raised to another power, multiply the exponents.

Correct answer: A.
m^3n^3

Explanation: By the power-of-a-product rule, (ab)^n=a^n b^n. Therefore, (mn)^3=m^3n^3. Option B represents a sum, while options C and D apply the third power to only one variable. Exam tip: when a power is outside a product in brackets, apply it to every factor.

Correct answer: A. \(\frac{p^2}{q^2}\)

Explanation: The exponent of a fraction applies to both numerator and denominator. Therefore \(\left(\frac{p}{q}\right)^2=\frac{p^2}{q^2}\).

Correct answer: C. \(4\cdot4\cdot4\)

Explanation: \(4^3\) means multiplying 4 by itself three times: \(4\times4\times4=64\). Therefore, option C is correct. Options A and B both equal 12, while option D equals \(3^4=81\). Exam tip: the exponent tells how many times the base is used as a factor; it is not multiplied directly by the base.

Correct answer: A. (25)

Explanation: The whole number ((-5)) is squared, so the answer is (25). Parentheses include the negative sign.

Correct answer: B. (-16)

Explanation: The direct answer is B: -16. The notation matters. In -4^2, there are no parentheses around -4, so exponentiation is performed before the leading minus sign. First calculate 4^2=16; then attach the minus sign: -(4^2)=-16. Option A, 16, would be correct for (-4)^2, where the negative sign is inside parentheses and the result is positive. Option B is correct for the expression exactly as written. Option C, 8, is unrelated because squaring means multiplying 4 by 4, not doubling it. Option D, -8, is also wrong because a square is not found by multiplying by 2. The key convention is that powers have priority over a separate leading sign. Thus -a^2 means -(a^2), while (-a)^2 means a^2. Confusing these two forms is a common exam mistake. Memory cue: no brackets, power first; brackets around the negative number, the whole negative base is squared.

Correct answer: B. \(20^2\)

Explanation: For two factors with the same exponent, use the law \(a^n\cdot b^n=(ab)^n\). Thus, \(4^2\cdot5^2=(4\cdot5)^2=20^2\). Option C is incorrect because the exponents are not added when the bases are multiplied in this form. Exam tip: Rewrite \(a^n b^n\) as \((ab)^n\) when the exponents are equal.

Correct answer: A. \(4^3\)

Explanation: With equal exponents, \(\frac{12^3}{3^3}=\left(\frac{12}{3}\right)^3=4^3\). Simplify equal powers together.

Correct answer: A. \(t^7\)

Explanation: When powers with the same base are multiplied, their exponents are added: \(t^m\cdot t^n=t^{m+n}\). Therefore, \(t^3\cdot t^4=t^{3+4}=t^7\). Option B incorrectly multiplies the exponents instead of adding them. Exam tip: In multiplication of powers with the same base, keep the base unchanged and add the exponents.

Correct answer: A. \(r^4\)

Explanation: When powers with the same non-zero base are divided, their exponents are subtracted: \(\frac{r^{10}}{r^6}=r^{10-6}=r^4\). Therefore, option A is correct. Option B results from adding the exponents, a rule used for multiplication rather than division. Exam tip: remember \(\frac{a^m}{a^n}=a^{m-n}\) for \(a\neq0\).

Correct answer: A. \(a^{-n}=\frac{1}{a^n}\), जहाँ \(a\ne0\)

Explanation: A negative exponent denotes the reciprocal of the corresponding positive power: \(a^{-n}=1/a^n\), provided \(a\ne0\). Option B only changes the sign, so it is incorrect. In exams, always check the non-zero base condition.