Operations on real numbers and the laws of exponents

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

\(4^3\) 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.

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

Without parentheses, the exponent is applied first, so (-4^2=-(4^2)=-16). Use parentheses for a negative base.

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.

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

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.

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 \(a\neq0\).

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.

The zero-exponent rule states that the zero power of every non-zero number is 1. Therefore, \(u^0=1\), \(v^0=1\), and \(2^0=1\). Hence, \(u^0+v^0+2^0=1+1+1=3\), so option C is correct. Option B may result from counting only two terms, but all three terms are equal to 1. Exam tip: apply \(a^0=1\) only when \(a\ne0\).

The negative-exponent law is \(x^{-n}=\frac{1}{x^n}\), where \(x\ne0\). Therefore, \(x^{-4}=\frac{1}{x^4}\), so option B is correct. Option A incorrectly drops the negative sign without taking the reciprocal, while option D treats the exponent 4 as a factor. Exam tip: For a negative exponent, write the base in the denominator and change the exponent to positive.