Operations on real numbers and the laws of exponents

Correct answer: A. (2^7)

Explanation: When the base is same, exponents are added, so (2^3\cdot2^4=2^{3+4}=2^7). In exams, add exponents for same-base multiplication.

Correct answer: B. (5^4)

Explanation: When the base is same, exponents are subtracted, so (\frac{5^6}{5^2}=5^{6-2}=5^4). For same-base division, subtract exponents.

Correct answer: B. (3^8)

Explanation: In a power of a power, exponents are multiplied, so ((3^2)^4=3^{2\cdot4}=3^8). Adding the exponents here is a common mistake.

Correct answer: B. 1

Explanation: By the law of exponents, the zero power of any non-zero number is 1: \(a^0=1\), where \(a\ne0\). Therefore, 1 is correct. The options \(a\) and \(-a\) depend on the base, while 0 is not the general value of a zero power. Exam tip: the zero power of every non-zero number is 1.

Correct answer: B. (\frac{1}{100})

Explanation: A negative exponent moves the number to the denominator, so (10^{-2}=\frac{1}{10^2}=\frac{1}{100}). A negative exponent does not mean a negative value.

Correct answer: A. (2^3\cdot5^3)

Explanation: The power of a product applies to both factors, so ((2\cdot5)^3=2^3\cdot5^3). Apply the outside exponent to every factor inside the bracket.

Correct answer: B. \(\frac{9}{16}\)

Explanation: The power of a fraction applies to both numerator and denominator, so \(\left(\frac{3}{4}\right)^2=\frac{3^2}{4^2}=\frac{9}{16}\). Do not square only the numerator.

Correct answer: A. \(7^5\)

Explanation: When powers with the same base are multiplied, their exponents are added: \(7^2\cdot 7^3=7^{2+3}=7^5\). Therefore, option A is correct. Option B incorrectly gives the exponent as 6. Exam tip: in multiplication, keep the common base unchanged and add the exponents.

Correct answer: A. (9^2)

Explanation: For division with the same base, exponents are subtracted, so (9^5\div9^3=9^{5-3}=9^2). Identify the law before calculating.

Correct answer: C. 4

Explanation: The governing exponent rule is a¹=a for every number a, including 4. An exponent of 1 means that the base occurs as one factor, so 4¹=4. Therefore option C is correct. It is useful to distinguish the distractors carefully: 4⁰=1, so option B would be correct only if the exponent were zero. Also, 4²=16, so option D corresponds to a square rather than a first power. Option A, 0, is not produced by raising the nonzero number 4 to the first power. The exponent does not mean that the base should be added, squared, or replaced; it specifies repeated multiplication, and one factor simply remains 4. Thus the expression has the exact value 4.

Correct answer: B. 9

Explanation: By the zero-exponent rule, the zeroth power of any non-zero number is 1, so \(6^0=1\). Also, \(2^3=8\). Therefore, \(6^0+2^3=1+8=9\), making option B correct. Exam tip: \(a^0=1\) applies when \(a\neq0\).

Correct answer: A. 25

Explanation: Using the order of operations, evaluate the powers first: \\(3^2=9\\) and \\(4^2=16\\). Their sum is \\(9+16=25\\), so the correct answer is 25. Option B, 49, is the square of 7, but the question asks for the sum of the squares of 3 and 4. Exam tip: calculate each exponent before performing the addition.

Correct answer: B. (a^{m-n})

Explanation: The direct answer is B: a^(m-n). When non-zero powers with the same base are divided, common factors cancel and the exponents are subtracted. For example, if m>n, a^m/a^n means a multiplied m times divided by a multiplied n times, leaving a^(m-n). The exponent law remains valid for all integer exponents when a≠0: a^m/a^n=a^(m-n). Option A, a^(m+n), is the rule for multiplication, not division. Option B is correct because division of like bases means subtract the denominator exponent from the numerator exponent. Option C, a^(mn), does not apply here; multiplying exponents is used in a power raised to another power, such as (a^m)^n=a^(mn). Option D, a^(n-m), reverses the subtraction and generally gives the reciprocal of the required result. The condition a≠0 matters because division by zero is undefined, and negative exponents may also occur. Memory cue: multiply like bases—add exponents; divide like bases—subtract exponents; power of a power—multiply exponents.

Correct answer: C. \(a^{mn}\)

Explanation: By the power-of-a-power law, the outer exponent \(n\) multiplies the inner exponent \(m\). Thus, \((a^m)^n=a^{mn}\), so option C is correct. Adding exponents, as in option A, applies when multiplying powers with the same base, not when raising one power to another. Exam tip: when a power is raised to another power, multiply the exponents.

Correct answer: A. x⁴y⁴

Explanation: The governing law is the power-of-a-product rule: (ab)ⁿ = aⁿbⁿ. The exponent 4 applies to the complete product xy, so (xy)⁴ means xy · xy · xy · xy. Regrouping the x factors and y factors gives x⁴y⁴. Therefore option A is correct. Option B raises only y to the fourth power and leaves x unchanged; option C does the reverse; and option D incorrectly changes multiplication into addition. It is important not to use the separate rule (a + b)ⁿ = aⁿ + bⁿ, which is generally false. Since x and y may be any suitable numbers, the symbolic product rule is the valid simplification.

Correct answer: A. \(\frac{x^3}{y^3}\)

Explanation: The exponent of a fraction applies to both numerator and denominator. Hence \(\left(\frac{x}{y}\right)^3=\frac{x^3}{y^3}\).

Correct answer: B. \(2 \times 2 \times 2 \times 2 \times 2\)

Explanation: \(2^5\) means multiplying 2 by itself five times: \(2 \times 2 \times 2 \times 2 \times 2\). Therefore, option B is correct. Option D represents adding 2 five times, which equals \(2 \times 5\), not \(2^5\). Exam tip: In \(a^n\), \(a\) is the base and \(n\) tells how many times the base is used as a factor.

Correct answer: A. (9)

Explanation: Here the whole number ((-3)) is squared, so ((-3)^2=9). Parentheses include the negative sign in the power.

Correct answer: B. (6^3)

Explanation: When exponents are the same, bases can be multiplied, so (2^3\cdot3^3=(2\cdot3)^3=6^3). This rule applies for equal exponents.

Correct answer: A. \(2^4\)

Explanation: With equal exponents, \(\frac{8^4}{4^4}=\left(\frac{8}{4}\right)^4=2^4\). Handling equal powers together is easier.

Correct answer: A. (x^7)

Explanation: For the same base (x), exponents are added, so (x^2\cdot x^5=x^7). The same law works for algebraic terms.

Correct answer: A. \(y^5\)

Explanation: When powers with the same base are divided, their exponents are subtracted: \(\frac{y^9}{y^4}=y^{9-4}=y^5\). Hence, option A is correct. Option D reverses the subtraction, while options B and C result from adding or multiplying the exponents. Exam tip: use \(\frac{a^m}{a^n}=a^{m-n}\) for \(a\ne0\).

Correct answer: B. \(p^6\)

Explanation: By the power-of-a-power law, 8a^m9^n = a^{mn}. Therefore, 8p^39^2 = p^{3\cdot2} = p^6, so option B is correct. Exam tip: when a power is raised to another power, multiply the exponents rather than adding them.

Correct answer: C. 2

Explanation: The zero power of every non-zero number is 1. Thus, \(m^0=1\) and \(n^0=1\), so \(m^0+n^0=1+1=2\). The option \(m+n\) is incorrect because the base does not remain in the result when its exponent is zero. Exam tip: remember that \(a^0=1\) for \(a\ne0\).

Correct answer: B. 1/a³

Explanation: The governing law for a negative integer exponent is a⁻ⁿ=1/aⁿ, provided a is nonzero. Applying this rule with n=3 gives a⁻³=1/a³, so option B is correct. The minus sign in the exponent means that the power is converted to its reciprocal; it does not place a minus sign before the whole expression. Thus option C is incorrect. Option A would be the result of simply ignoring the negative exponent and losing the reciprocal. Option D incorrectly treats 3 as a coefficient in the denominator, whereas the exponent applies to a and produces a³. The condition a≠0 is essential because a³ appears in the denominator, and division by zero is undefined. Therefore the exact equivalent form is 1/a³.